Topology glossary

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This is a glossary of some terms used in the branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
known as
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. Although there is no absolute distinction between different areas of topology, the focus here is on
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geome ...
. The following definitions are also fundamental to
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
,
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
and
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated ...
. All spaces in this glossary are assumed to be
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s unless stated otherwise.

# A

;Absolutely closed: See ''H-closed'' ;Accessible: See $T_1$. ;Accumulation point: See
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
. ; Alexandrov topology: The topology of a space ''X'' is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the
upper set In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
s of a
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
. ;Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces. ;α-closed, α-open: A subset ''A'' of a topological space ''X'' is α-open if $A \subseteq \operatorname_X \left\left( \operatorname_X \left\left( \operatorname_X A \right\right) \right\right)$, and the complement of such a set is α-closed. ; Approach space: An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.

# B

;Baire space: This has two distinct common meanings: :#A space is a Baire space if the intersection of any
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
collection of dense open sets is dense; see
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
. :#Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see
Baire space (set theory) In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called "reals". It is denoted NN, ...
. ; Base: A collection ''B'' of open sets is a base (or basis) for a topology $\tau$ if every open set in $\tau$ is a union of sets in $B$. The topology $\tau$ is the smallest topology on $X$ containing $B$ and is said to be generated by $B$. ; Basis: See Base. ;β-open: See ''Semi-preopen''. ;b-open, b-closed: A subset $A$ of a topological space $X$ is b-open if $A \subseteq \operatorname_X \left\left( \operatorname_X A \right\right) \cup \operatorname_X \left\left( \operatorname_X A \right\right)$. The complement of a b-open set is b-closed. ;
Borel algebra In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
: The
Borel algebra In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
on a topological space $\left(X,\tau\right)$ is the smallest $\sigma$-algebra containing all the open sets. It is obtained by taking intersection of all $\sigma$-algebras on $X$ containing $\tau$. ;Borel set: A Borel set is an element of a Borel algebra. ;
Boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
: The
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
(or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set $A$ is denoted by $\partial A$ or $bd$ $A$. ; Bounded: A set in a metric space is bounded if it has
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
taking values in a metric space is bounded if its
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
is a bounded set.

# C

;
Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again conti ...
: The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
Top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few ...
has
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s as objects and
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s as
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
s. ;
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
: A
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
in a metric space (''M'', ''d'') is a
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
if, for every positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
''r'', there is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''''m'', ''x''''n'') < ''r''. ;
Clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
: A set is
clopen In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
if it is both open and closed. ;Closed ball: If (''M'', ''d'') is a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, a closed ball is a set of the form ''D''(''x''; ''r'') := , where ''x'' is in ''M'' and ''r'' is a positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
, the radius of the ball. A closed ball of radius ''r'' is a closed ''r''-ball. Every closed ball is a closed set in the topology induced on ''M'' by ''d''. Note that the closed ball ''D''(''x''; ''r'') might not be equal to the closure of the open ball ''B''(''x''; ''r''). ;
Closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
: A set is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
if its complement is a member of the topology. ; Closed function: A function from one space to another is closed if the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
of every closed set is closed. ; Closure: The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set ''S'' is a point of closure of ''S''. ;Closure operator: See Kuratowski closure axioms. ; Coarser topology: If ''X'' is a set, and if ''T''1 and ''T''2 are topologies on ''X'', then ''T''1 is coarser (or smaller, weaker) than ''T''2 if ''T''1 is contained in ''T''2. Beware, some authors, especially analysts, use the term stronger. ;Comeagre: A subset ''A'' of a space ''X'' is comeagre (comeager) if its
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
''X''\''A'' is meagre. Also called residual. ;
Compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
: A space is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
if every open cover has a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
Compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory an ...
: The
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory an ...
on the set ''C''(''X'', ''Y'') of all continuous maps between two spaces ''X'' and ''Y'' is defined as follows: given a compact subset ''K'' of ''X'' and an open subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all maps ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a subbase for the compact-open topology. ; Complete: A metric space is complete if every Cauchy sequence converges. ;Completely metrizable/completely metrisable: See
complete space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. ;Completely normal: A space is completely normal if any two separated sets have disjoint neighbourhoods. ;Completely normal Hausdorff: A completely normal Hausdorff space (or T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is T1, so the terminology is
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
.) Every completely normal Hausdorff space is normal Hausdorff. ;
Completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
: A space is
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and are functionally separated. ; Completely T3: See Tychonoff. ;Component: See Connected component/Path-connected component. ; Connected: A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set. ; Connected component: A connected component of a space is a maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a partition of that space. ; Continuous: A function from one space to another is continuous if the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of every open set is open. ; Continuum: A space is called a continuum if it a compact, connected Hausdorff space. ;
Contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within tha ...
: A space ''X'' is contractible if the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
on ''X'' is homotopic to a constant map. Every contractible space is simply connected. ; Coproduct topology: If is a collection of spaces and ''X'' is the (set-theoretic)
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of , then the coproduct topology (or disjoint union topology, topological sum of the ''X''''i'') on ''X'' is the finest topology for which all the injection maps are continuous. ; Cosmic space: A continuous
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
of some separable
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. ; Countable chain condition: A space ''X'' satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable. ; Countably compact: A space is countably compact if every
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
open cover has a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
subcover. Every countably compact space is pseudocompact and weakly countably compact. ;Countably locally finite: A collection of subsets of a space ''X'' is countably locally finite (or σ-locally finite) if it is the union of a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
collection of locally finite collections of subsets of ''X''. ; Cover: A collection of subsets of a space is a cover (or covering) of that space if the union of the collection is the whole space. ;Covering: See Cover. ;Cut point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a cut point if the subspace ''X'' − is disconnected.

# D

;δ-cluster point, δ-closed, δ-open: A point ''x'' of a topological space ''X'' is a δ-cluster point of a subset ''A'' if $A \cap \operatorname_X\left\left( \operatorname_X\left(U\right) \right\right) \neq \emptyset$ for every open neighborhood ''U'' of ''x'' in ''X''. The subset ''A'' is δ-closed if it is equal to the set of its δ-cluster points, and δ-open if its complement is δ-closed. ;
Dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
: A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space. ; Dense-in-itself set: A set is dense-in-itself if it has no
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
. ;Density: the minimal cardinality of a dense subset of a topological space. A set of density ℵ0 is a
separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of t ...
. ;Derived set: If ''X'' is a space and ''S'' is a subset of ''X'', the derived set of ''S'' in ''X'' is the set of limit points of ''S'' in ''X''. ;Developable space: A topological space with a development. ; Development: A
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
collection of
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
s of a topological space, such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is disjoint from ''C''. ;Diameter: If (''M'', ''d'') is a metric space and ''S'' is a subset of ''M'', the diameter of ''S'' is the
supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of the distances ''d''(''x'', ''y''), where ''x'' and ''y'' range over ''S''. ;Discrete metric: The discrete metric on a set ''X'' is the function ''d'' : ''X'' × ''X''  →  R such that for all ''x'', ''y'' in ''X'', ''d''(''x'', ''x'') = 0 and ''d''(''x'', ''y'') = 1 if ''x'' ≠ ''y''. The discrete metric induces the discrete topology on ''X''. ;
Discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
: A space ''X'' is
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
if every subset of ''X'' is open. We say that ''X'' carries the discrete topology.Steen & Seebach (1978) p.41 ;
Discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest t ...
: See
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
. ;Disjoint union topology: See Coproduct topology. ; Dispersion point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a dispersion point if the subspace ''X'' − is hereditarily disconnected (its only connected components are the one-point sets). ;Distance: See
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. ; Dunce hat (topology)

# E

; Entourage: See
Uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
. ;Exterior: The exterior of a set is the interior of its complement.

# F

; ''F''σ set: An ''F''σ set is a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
union of closed sets. ;
Filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
Filters in topology Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some gi ...
. A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold: :# The
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is not in ''F''. :# The intersection of any
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
number of elements of ''F'' is again in ''F''. :# If ''A'' is in ''F'' and if ''B'' contains ''A'', then ''B'' is in ''F''. ;
Final topology In general topology and related areas of mathematics, the final topology (or coinduced, strong, colimit, or inductive topology) on a set X, with respect to a family of functions from topological spaces into X, is the finest topology on X that m ...
: On a set ''X'' with respect to a family of functions into $X$, is the finest topology on ''X'' which makes those functions continuous. ;
Fine topology (potential theory) In mathematics, in the field of potential theory, the fine topology is a natural topology for setting the study of subharmonic functions. In the earliest studies of subharmonic functions, namely those for which \Delta u \ge 0, where \Delta is th ...
: On
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
$\R^n$, the coarsest topology making all
subharmonic function In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functi ...
s (equivalently all superharmonic functions) continuous. ;
Finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
: If ''X'' is a set, and if ''T''1 and ''T''2 are topologies on ''X'', then ''T''2 is finer (or larger, stronger) than ''T''1 if ''T''2 contains ''T''1. Beware, some authors, especially analysts, use the term weaker. ;Finitely generated: See Alexandrov topology. ; First category: See Meagre. ;
First-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
: A space is
first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
if every point has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
local base. ;Fréchet: See T1. ;Frontier: See
Boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
. ;Full set: A
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
subset ''K'' of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by t ...
is called full if its
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
is connected. For example, the closed unit disk is full, while the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
is not. ;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X''  →
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
such that ''f''(''A'') = 0 and ''f''(''B'') = 1.

# G

; ''G''δ set: A ''G''δ set or inner limiting set is a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
intersection of open sets.Steen & Seebach (1978) p.162 ;''G''δ space: A space in which every closed set is a ''G''δ set. ;
Generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
: A
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
for a closed set is a point for which the closed set is the closure of the singleton set containing that point.

# H

; Hausdorff: A
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
(or T2 space) is one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is T1. ; H-closed: A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff space containing it. ; Hereditarily ''P'': A space is hereditarily ''P'' for some property ''P'' if every subspace is also ''P''. ;
Hereditary Heredity, also called inheritance or biological inheritance, is the passing on of traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cells or organisms acquire the genetic info ...
: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.Steen & Seebach p.4 For example, second-countability is a hereditary property. ;
Homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
: If ''X'' and ''Y'' are spaces, a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomo ...
from ''X'' to ''Y'' is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''−1 are continuous. The spaces ''X'' and ''Y'' are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical. ;
Homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size ...
: A space ''X'' is
homogeneous Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, size ...
if, for every ''x'' and ''y'' in ''X'', there is a homeomorphism ''f'' : ''X''  →  ''X'' such that ''f''(''x'') = ''y''. Intuitively, the space looks the same at every point. Every
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
is homogeneous. ; Homotopic maps: Two continuous maps ''f'', ''g'' : ''X''  →  ''Y'' are
homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
(in ''Y'') if there is a continuous map ''H'' : ''X'' ×
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
→  ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' ×
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
is given the product topology. The function ''H'' is called a homotopy (in ''Y'') between ''f'' and ''g''. ; Homotopy: See Homotopic maps. ; Hyper-connected: A space is hyper-connected if no two non-empty open sets are disjoint Every hyper-connected space is connected.

# I

; Identification map: See
Quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
. ;
Identification space In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient ...
: See Quotient space. ;
Indiscrete space In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
: See
Trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
. ; Infinite-dimensional topology: See
Hilbert manifold In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold prov ...
and Q-manifolds, i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively. ; Inner limiting set: A ''G''δ set. ;
Interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
: The
interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set ''S'' is an interior point of ''S''. ; Interior point: See
Interior Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior de ...
. ;
Isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
: A point ''x'' is an
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
if the singleton (mathematics), singleton is open. More generally, if ''S'' is a subset of a space ''X'', and if ''x'' is a point of ''S'', then ''x'' is an isolated point of ''S'' if is open in the subspace topology on ''S''. ; Isometric isomorphism: If ''M''1 and ''M''2 are metric spaces, an isometric isomorphism from ''M''1 to ''M''2 is a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
isometry ''f'' : ''M''1  →  ''M''2. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical. ; Isometry: If (''M''1, ''d''1) and (''M''2, ''d''2) are metric spaces, an isometry from ''M''1 to ''M''2 is a function ''f'' : ''M''1  →  ''M''2 such that ''d''2(''f''(''x''), ''f''(''y'')) = ''d''1(''x'', ''y'') for all ''x'', ''y'' in ''M''1. Every isometry is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
, although not every isometry is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
.

# K

; Kolmogorov axiom: See T0. ; Kuratowski closure axioms: The Kuratowski closure axioms is a set of
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s satisfied by the function which takes each subset of ''X'' to its closure: :# ''
Isotonicity In chemical biology, tonicity is a measure of the effective osmotic pressure gradient; the water potential of two solutions separated by a partially-permeable cell membrane. Tonicity depends on the relative concentration of selective membrane- ...
'': Every set is contained in its closure. :# ''
Idempotence Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
'': The closure of the closure of a set is equal to the closure of that set. :# ''Preservation of binary unions'': The closure of the union of two sets is the union of their closures. :# ''Preservation of nullary unions'': The closure of the empty set is empty. :If ''c'' is a function from the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of ''X'' to itself, then ''c'' is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on ''X'' by declaring the closed sets to be the fixed points of this operator, i.e. a set ''A'' is closed
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''c''(''A'') = ''A''. ;Kolmogorov topology :T''Kol'' = ∪; the pair (R,T''Kol'') is named ''Kolmogorov Straight''.

# L

; L-space: An ''L-space'' is a hereditarily
Lindelöf space In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of ''compactness'', which requires the existence of a ''finite'' sub ...
which is not hereditarily separable. A Suslin line would be an L-space. ;Larger topology: See
Finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
. ;
Limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
: A point ''x'' in a space ''X'' is a
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also cont ...
of a subset ''S'' if every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself. ;Limit point compact: See Weakly countably compact. ; Lindelöf: A space is Lindelöf if every open cover has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
subcover. ;
Local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
: A set ''B'' of neighbourhoods of a point ''x'' of a space ''X'' is a local base (or local basis, neighbourhood base, neighbourhood basis) at ''x'' if every neighbourhood of ''x'' contains some member of ''B''. ;Local basis: See Local base. ;Locally (P) space: There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P). The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected.Hart et al (2004) p.65 ; Locally closed subset: A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure. ;
Locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
: A space is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ...
if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces. Every locally compact Hausdorff space is Tychonoff. ;
Locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
: A space is
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
if every point has a local base consisting of connected neighbourhoods. ; Locally dense: see ''Preopen''. ; Locally finite: A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
ly many of the subsets. See also countably locally finite, point finite. ;Locally metrizable/Locally metrisable: A space is locally metrizable if every point has a metrizable neighbourhood. ;
Locally path-connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectednes ...
: A space is
locally path-connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectednes ...
if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is path-connected. ;
Locally simply connected In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locall ...
: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods. ; Loop: If ''x'' is a point in a space ''X'', a loop at ''x'' in ''X'' (or a loop in ''X'' with basepoint ''x'') is a path ''f'' in ''X'', such that ''f''(0) = ''f''(1) = ''x''. Equivalently, a loop in ''X'' is a continuous map from the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
''S''1 into ''X''.

# M

; Meagre: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is meagre in ''X'' (or of first category in ''X'') if it is the
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
union of nowhere dense sets. If ''A'' is not meagre in ''X'', ''A'' is of second category in ''X''.Steen & Seebach (1978) p.7 ; Metacompact: A space is metacompact if every open cover has a point finite open refinement. ;Metric: See
Metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
. ;Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism. ; Metric map: If ''X'' and ''Y'' are metric spaces with metrics ''d''''X'' and ''d''''Y'' respectively, then a metric map is a function ''f'' from ''X'' to ''Y'', such that for any points ''x'' and ''y'' in ''X'', ''d''''Y''(''f''(''x''), ''f''(''y'')) ≤ ''d''''X''(''x'', ''y''). A metric map is strictly metric if the above inequality is strict for all ''x'' and ''y'' in ''X''. ;
Metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
: A
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
(''M'', ''d'') is a set ''M'' equipped with a function ''d'' : ''M'' × ''M'' →  R satisfying the following axioms for all ''x'', ''y'', and ''z'' in ''M'': :# ''d''(''x'', ''y'') ≥ 0 :# ''d''(''x'', ''x'') = 0 :# if   ''d''(''x'', ''y'') = 0   then   ''x'' = ''y''     (''identity of indiscernibles'') :# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')     (''symmetry'') :# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'')     (''
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
'') :The function ''d'' is a metric on ''M'', and ''d''(''x'', ''y'') is the distance between ''x'' and ''y''. The collection of all open balls of ''M'' is a base for a topology on ''M''; this is the topology on ''M'' induced by ''d''. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. ;
Metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
/Metrisable: A space is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable. ;Monolith: Every non-empty ultra-connected compact space ''X'' has a largest proper open subset; this subset is called a monolith. ; Moore space: A Moore space is a developable regular Hausdorff space.Steen & Seebach (1978) p.163

# N

; Nearly open: see ''preopen''. ;
Neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
/Neighborhood: A neighbourhood of a point ''x'' is a set containing an open set which in turn contains the point ''x''. More generally, a neighbourhood of a set ''S'' is a set containing an open set which in turn contains the set ''S''. A neighbourhood of a point ''x'' is thus a neighbourhood of the singleton (mathematics), singleton set . (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.) ; Neighbourhood base/basis: See
Local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbou ...
. ;Neighbourhood system for a point ''x'': A
neighbourhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
at a point ''x'' in a space is the collection of all neighbourhoods of ''x''. ; Net: A net in a space ''X'' is a map from a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
''A'' to ''X''. A net from ''A'' to ''X'' is usually denoted (''x''α), where α is an index variable ranging over ''A''. Every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
is a net, taking ''A'' to be the directed set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s with the usual ordering. ; Normal: A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are ...
. ; Normal Hausdorff: A normal Hausdorff space (or T4 space) is a normal T1 space. (A normal space is Hausdorff
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff. ;
Nowhere dense In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
: A nowhere dense set is a set whose closure has empty interior.

# O

;
Open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
: An
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...
is a cover consisting of open sets. ; Open ball: If (''M'', ''d'') is a metric space, an open ball is a set of the form ''B''(''x''; ''r'') := , where ''x'' is in ''M'' and ''r'' is a positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
, the radius of the ball. An open ball of radius ''r'' is an open ''r''-ball. Every open ball is an open set in the topology on ''M'' induced by ''d''. ; Open condition: See open property. ;
Open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
: An
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
is a member of the topology. ; Open function: A function from one space to another is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
if the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
of every open set is open. ; Open property: A property of points in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
is said to be "open" if those points which possess it form an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
. Such conditions often take a common form, and that form can be said to be an ''open condition''; for example, in
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s, one defines an open ball as above, and says that "strict inequality is an open condition".

# P

;
Paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
: A space is
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is norma ...
if every open cover has a locally finite open refinement. Paracompact implies metacompact.Steen & Seebach (1978) p.23 Paracompact Hausdorff spaces are normal.Steen & Seebach (1978) p.25 ;
Partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are ...
: A partition of unity of a space ''X'' is a set of continuous functions from ''X'' to
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
such that any point has a neighbourhood where all but a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. ;
Path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire ...
: A
path A path is a route for physical travel – see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire ...
in a space ''X'' is a continuous map ''f'' from the closed unit interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
into ''X''. The point ''f''(0) is the initial point of ''f''; the point ''f''(1) is the terminal point of ''f''.Steen & Seebach (1978) p.29 ;
Path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
: A space ''X'' is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
if, for every two points ''x'', ''y'' in ''X'', there is a path ''f'' from ''x'' to ''y'', i.e., a path with initial point ''f''(0) = ''x'' and terminal point ''f''(1) = ''y''. Every path-connected space is connected. ;Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a partition of that space, which is finer than the partition into connected components. The set of path-connected components of a space ''X'' is denoted π0(''X''). ;Perfectly normal: a normal space which is also a Gδ. ;π-base: A collection ''B'' of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from ''B''. ;Point: A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point". ;Point of closure: See Closure. ;
Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken *Polish brothers (Mark Polish and Michael Polish, born 1970), American twin scree ...
: A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space. ; Polyadic: A space is polyadic if it is the continuous image of the power of a one-point compactification of a locally compact, non-compact Hausdorff space. ;P-point: A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections. ;Pre-compact: See
Relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
. ;: A subset ''A'' of a topological space ''X'' is preopen if $A \subseteq \operatorname_X \left\left( \operatorname_X A \right\right)$. ;Prodiscrete topology: The prodiscrete topology on a product ''A''''G'' is the product topology when each factor ''A'' is given the discrete topology. ;
Product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
: If $\left\left(X_i\right\right)$ is a collection of spaces and ''X'' is the (set-theoretic)
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
of $\left\left(X_i\right\right),$ then the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
on ''X'' is the coarsest topology for which all the projection maps are continuous. ;Proper function/mapping: A continuous function ''f'' from a space ''X'' to a space ''Y'' is proper if $f^\left(C\right)$ is a compact set in ''X'' for any compact subspace ''C'' of ''Y''. ;
Proximity space In topology, a proximity space, also called a nearness space, is an axiomatization of the intuitive notion of "nearness" that hold set-to-set, as opposed to the better known point-to-set notion that characterize topological spaces. The concept was ...
: A proximity space (''X'', d) is a set ''X'' equipped with a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
d between subsets of ''X'' satisfying the following properties: :For all subsets ''A'', ''B'' and ''C'' of ''X'', :#''A'' d ''B'' implies ''B'' d ''A'' :#''A'' d ''B'' implies ''A'' is non-empty :#If ''A'' and ''B'' have non-empty intersection, then ''A'' d ''B'' :#''A'' d (''B'' $\cup$ ''C'')
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
(''A'' d ''B'' or ''A'' d ''C'') :#If, for all subsets ''E'' of ''X'', we have (''A'' d ''E'' or ''B'' d ''E''), then we must have ''A'' d (''X'' − ''B'') ; Pseudocompact: A space is pseudocompact if every real-valued continuous function on the space is bounded. ;Pseudometric: See Pseudometric space. ;
Pseudometric space In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metri ...
: A pseudometric space (''M'', ''d'') is a set ''M'' equipped with a real-valued function $d : M \times M \to \R$ satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function ''d'' is a pseudometric on ''M''. Every metric is a pseudometric. ;Punctured neighbourhood/Punctured neighborhood: A punctured neighbourhood of a point ''x'' is a neighbourhood of ''x'', minus . For instance, the interval (−1, 1) = is a neighbourhood of ''x'' = 0 in the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
, so the set $\left(-1, 0\right) \cup \left(0, 1\right) = \left(-1, 1\right) - \$ is a punctured neighbourhood of 0.

# Q

;Quasicompact: See
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. Some authors define "compact" to include the Hausdorff separation axiom, and they use the term quasicompact to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French. ;Quotient map: If ''X'' and ''Y'' are spaces, and if ''f'' is a
surjection In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
from ''X'' to ''Y'', then ''f'' is a quotient map (or identification map) if, for every subset ''U'' of ''Y'', ''U'' is open in ''Y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''f'' 1(''U'') is open in ''X''. In other words, ''Y'' has the ''f''-strong topology. Equivalently, $f$ is a quotient map if and only if it is the transfinite composition of maps $X\rightarrow X/Z$, where $Z\subset X$ is a subset. Note that this does not imply that ''f'' is an open function. ; Quotient space: If ''X'' is a space, ''Y'' is a set, and ''f'' : ''X'' → ''Y'' is any
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
function, then the
Quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
on ''Y'' induced by ''f'' is the finest topology for which ''f'' is continuous. The space ''X'' is a quotient space or identification space. By definition, ''f'' is a quotient map. The most common example of this is to consider an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on ''X'', with ''Y'' the set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es and ''f'' the natural projection map. This construction is dual to the construction of the subspace topology.

# R

; Refinement: A cover ''K'' is a refinement of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''. ; Regular: A space is regular if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have disjoint neighbourhoods. ; Regular Hausdorff: A space is regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorff
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is T0, so the terminology is consistent.) ; : A subset of a space ''X'' is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior.Steen & Seebach (1978) p.6 An example of a non-regular open set is the set ''U'' = ∪ in R with its normal topology, since 1 is in the interior of the closure of ''U'', but not in ''U''. The regular open subsets of a space form a
complete Boolean algebra In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boole ...
. ;
Relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
: A subset ''Y'' of a space ''X'' is
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
in ''X'' if the closure of ''Y'' in ''X'' is compact. ; Residual: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is residual in ''X'' if the complement of ''A'' is meagre in ''X''. Also called comeagre or comeager. ; Resolvable: A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
is called resolvable if it is expressible as the union of two disjoint
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
s. ; Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.

# S

; S-space: An ''S-space'' is a hereditarily
separable space In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of t ...
which is not hereditarily Lindelöf. ; Scattered: A space ''X'' is scattered if every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''. ; Scott: The Scott topology on a
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
is that in which the open sets are those
Upper set In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a partially ordered set (X, \leq) is a subset S \subseteq X with the following property: if ''s'' is in ''S'' and if ''x'' in ''X'' is larger ...
s inaccessible by directed joins. ;Second category: See Meagre. ;
Second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
: A space is
second-countable In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
or perfectly separable if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
base for its topology. Every second-countable space is first-countable, separable, and Lindelöf. ;
Semilocally simply connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected if ...
: A space ''X'' is
semilocally simply connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space ''X'' is semi-locally simply connected if ...
if, for every point ''x'' in ''X'', there is a neighbourhood ''U'' of ''x'' such that every loop at ''x'' in ''U'' is homotopic in ''X'' to the constant loop ''x''. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in ''X'', whereas in the definition of locally simply connected, the homotopy must live in ''U''.) ;Semi-open: A subset ''A'' of a topological space ''X'' is called semi-open if $A \subseteq \operatorname_X \left\left( \operatorname_X A \right\right)$. ;Semi-preopen: A subset ''A'' of a topological space ''X'' is called semi-preopen if $A \subseteq \operatorname_X \left\left( \operatorname_X \left\left( \operatorname_X A \right\right) \right\right)$ ; Semiregular: A space is semiregular if the regular open sets form a base. ; Separable: A space is separable if it has a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
dense subset. ; Separated: Two sets ''A'' and ''B'' are separated if each is disjoint from the other's closure. ;
Sequentially compact In mathematics, a topological space ''X'' is sequentially compact if every sequence of points in ''X'' has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notio ...
: A space is sequentially compact if every
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact. ;
Short map In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 19 ...
: See metric map ;
Simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
: A space is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
if it is path-connected and every loop is homotopic to a constant map. ;Smaller topology: See Coarser topology. ; Sober: In a
sober space In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point. Definitio ...
, every irreducible closed subset is the closure of exactly one point: that is, has a unique
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
. ;Star: The star of a point in a given cover of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
is the union of all the sets in the cover that contain the point. See
star refinement In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''. The general definition makes sense for arbitrary coverings and doe ...
. ;$f$-Strong topology: Let $f\colon X\rightarrow Y$ be a map of topological spaces. We say that $Y$ has the $f$-strong topology if, for every subset $U\subset Y$, one has that $U$ is open in $Y$ if and only if $f^\left(U\right)$ is open in $X$ ;Stronger topology: See
Finer topology In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the ...
. Beware, some authors, especially analysts, use the term weaker topology. ;
Subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
: A collection of open sets is a
subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
(or subbasis) for a topology if every non-empty proper open set in the topology is a union of
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
intersections of sets in the subbase. If ''B'' is ''any'' collection of subsets of a set ''X'', the topology on ''X'' generated by ''B'' is the smallest topology containing ''B''; this topology consists of the empty set, ''X'' and all unions of finite intersections of elements of ''B''. ; Subbasis: See
Subbase In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
. ;Subcover: A cover ''K'' is a subcover (or subcovering) of a cover ''L'' if every member of ''K'' is a member of ''L''. ;Subcovering: See Subcover. ; Submaximal space: A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are s ...
and a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
. Here are some facts about submaximality as a property of topological spaces: * Every door space is submaximal. * Every submaximal space is ''weakly submaximal'' viz every finite set is locally closed. * Every submaximal space is irresolvable. ;Subspace: If ''T'' is a topology on a space ''X'', and if ''A'' is a subset of ''X'', then the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
on ''A'' induced by ''T'' consists of all intersections of open sets in ''T'' with ''A''. This construction is dual to the construction of the quotient topology.

# T

; T0: A space is T0 (or Kolmogorov) if for every pair of distinct points ''x'' and ''y'' in the space, either there is an open set containing ''x'' but not ''y'', or there is an open set containing ''y'' but not ''x''. ; T1: A space is T1 (or Fréchet or accessible) if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singleton (mathematics), singletons are closed. Every T1 space is T0. ; T2: See
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
. ; T3: See Regular Hausdorff. ; T: See
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...
. ; T4: See Normal Hausdorff. ; T5: See Completely normal Hausdorff. ;
Top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few ...
: See
Category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again conti ...
. ;θ-cluster point, θ-closed, θ-open: A point ''x'' of a topological space ''X'' is a θ-cluster point of a subset ''A'' if $A \cap \operatorname_X\left(U\right) \neq \emptyset$ for every open neighborhood ''U'' of ''x'' in ''X''. The subset ''A'' is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed. ;
Topological invariant In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
: A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not.
Algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
is the study of topologically invariant
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
constructions on topological spaces. ;
Topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
: A
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
(''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s: :# The empty set and ''X'' are in ''T''. :# The union of any collection of sets in ''T'' is also in ''T''. :# The intersection of any pair of sets in ''T'' is also in ''T''. :The collection ''T'' is a topology on ''X''. ;Topological sum: See Coproduct topology. ;Topologically complete: Completely metrizable spaces (i. e. topological spaces homeomorphic to complete metric spaces) are often called ''topologically complete''; sometimes the term is also used for Čech-complete spaces or
completely uniformizable space In mathematics, a topological space (''X'', ''T'') is called completely uniformizable (or Dieudonné complete) if there exists at least one complete uniformity that induces the topology ''T''. Some authors additionally require ''X'' to be Hausdorf ...
s. ;Topology: See
Topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
. ;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
cover of ''M'' by open balls of radius ''r''. A metric space is compact if and only if it is complete and totally bounded. ;Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point. ;
Trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
: The
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...
(or indiscrete topology) on a set ''X'' consists of precisely the empty set and the entire space ''X''. ; Tychonoff: A
Tychonoff space In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that i ...
(or completely regular Hausdorff space, completely T3 space, T3.5 space) is a completely regular T0 space. (A completely regular space is Hausdorff
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
it is T0, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

# U

;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint. Every ultra-connected space is path-connected. ; Ultrametric: A metric is an ultrametric if it satisfies the following stronger version of the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
: for all ''x'', ''y'', ''z'' in ''M'', ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')). ; Uniform isomorphism: If ''X'' and ''Y'' are
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
s, a uniform isomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''−1 are
uniformly continuous In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
. The spaces are then said to be uniformly isomorphic and share the same uniform properties. ; Uniformizable/Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space. ;
Uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
: A
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
is a set ''X'' equipped with a nonempty collection Φ of subsets of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
''X'' × ''X'' satisfying the following
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s: :# if ''U'' is in Φ, then ''U'' contains . :# if ''U'' is in Φ, then is also in Φ :# if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ :# if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ :# if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''. :The elements of Φ are called entourages, and Φ itself is called a uniform structure on ''X''. The uniform structure induces a topology on ''X'' where the basic neighborhoods of ''x'' are sets of the form for ''U''∈Φ. ;Uniform structure: See
Uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
.

# W

;
Weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
: The
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous. ; Weaker topology: See Coarser topology. Beware, some authors, especially analysts, use the term stronger topology. ; Weakly countably compact: A space is weakly countably compact (or limit point compact) if every
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
subset has a limit point. ; Weakly hereditary: A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary. ; Weight: The weight of a space ''X'' is the smallest
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
κ such that ''X'' has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is
well-ordered In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
.) ; Well-connected: See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

# Z

;Zero-dimensional: A space is
zero-dimensional In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical i ...
if it has a base of clopen sets.Steen & Seebach (1978) p.33

*
Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It d ...
,
Axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, and
Function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
for definitions concerning sets and functions. *
Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
for a brief history and description of the subject area *
Topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s for basic definitions and examples * List of general topology topics * List of examples in general topology ;Topology specific concepts *
Compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
*
Connected space In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
* Continuity *
Metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
*
Separated sets In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets a ...
* Separation axiom *
Topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
*
Uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and un ...
;Other glossaries * Glossary of algebraic topology *
Glossary of differential geometry and topology This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: * Glossary of general topology * Glossary of algebraic topology * Glossary of Riemannian and metric geom ...
* Glossary of areas of mathematics *
Glossary of Riemannian and metric geometry This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology. The following articles may also be useful; they either contain specialised vocabulary or provid ...

# References

* * * * * * * Also available as Dover reprint.