punctured plane

TheInfoList

This is a glossary of some terms used in the branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
known as
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

. Although there is no absolute distinction between different areas of topology, the focus here is on
general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is t ...
. 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
,
differential topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and
geometric topology In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. All spaces in this glossary are assumed to be
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s unless stated otherwise.

# A

;Absolutely closed: See ''H-closed'' ;Accessible: See $T_1$. ;Accumulation point: See
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
. ;
Alexandrov topologyIn topology, an Alexandrov topology is a topological space, topology in which the intersection (set theory), intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is op ...
: The topology of a space ''X'' is an
Alexandrov topologyIn topology, an Alexandrov topology is a topological space, topology in which the intersection (set theory), intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is op ...
(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 Image:Upset.svg, A Hasse diagram of the power set of the set with the upper set ↑ colored green. The white sets form the lower set ↓. In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in ''X'') of a par ...
s of a
poset 250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable. In mathematics, especially order the ...
. ;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 spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
: An
approach spaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
collection of dense open sets is dense; see
Baire space In mathematics, a Baire space is a topological space such that every intersection of a countable collection of Open set, open dense sets in the space is also dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire ...
. :#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 (mathematics), 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 ...
. ;
Base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
: A collection ''B'' of open sets is a
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
(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 Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
: See
Base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
. ;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 algebraIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: The
Borel algebraIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
: The
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
(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 Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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 function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
taking values in a metric space is bounded if its
image An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows ...
is a bounded set.

# C

;Category of topological spaces: The category theory, category Category of topological spaces, Top has
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s as object (category theory), objects and continuous maps as morphisms. ;Cauchy sequence: A sequence in a metric space (''M'', ''d'') is a Cauchy sequence if, for every positive number, positive real number ''r'', there is an integer ''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''''m'', ''x''''n'') < ''r''. ;Clopen set: A set is clopen set, clopen if it is both open and closed. ;Closed ball: If (''M'', ''d'') is a metric space, a closed ball is a set of the form ''D''(''x''; ''r'') := , where ''x'' is in ''M'' and ''r'' is a positive number, positive real number, 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 (topology), closure of the open ball ''B''(''x''; ''r''). ;Closed set: A set is Closed set, closed if its complement is a member of the topology. ;Closed function: A function from one space to another is closed if the image (mathematics), image of every closed set is closed. ;Closure (topology), Closure: The closure (topology), 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 topology, coarser (or smaller, weaker) than ''T''2 if ''T''1 is contained in ''T''2. Beware, some authors, especially mathematical analysis, analysts, use the term stronger. ;Comeagre: A subset ''A'' of a space ''X'' is comeagre (comeager) if its complement (set theory), complement ''X''\''A'' is meagre set, meagre. Also called residual. ;Compact space, Compact: A space is compact space, compact if every open cover has a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact. ;Compact-open topology: The compact-open topology 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 space, Complete: A metric space is complete space, complete if every Cauchy sequence converges. ;Completely metrizable/completely metrisable: See complete space. ;Completely normal: A space is completely normal if any two separated sets have Disjoint sets, disjoint neighbourhoods. ;Completely normal Hausdorff: A completely normal Hausdorff space (or T5 space, T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff. ;Completely regular space, Completely regular: A space is Completely regular space, completely regular if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and are functionally separated. ;Completely T3 space, Completely T3: See Tychonoff space, Tychonoff. ;Component: See Connected space, Connected component/Path-connected component. ;Connected (topology), Connected: A space is connected (topology), connected if it is not the union of a pair of Disjoint sets, disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set. ;connected space, Connected component: A connected space, connected component of a space is a maximal set, maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a partition of a set, partition of that space. ;Continuity (topology), Continuous: A function from one space to another is continuity (topology), continuous if the preimage of every open set is open. ;Continuum: A space is called a continuum if it a compact, connected Hausdorff space. ;Contractible space, Contractible: A space ''X'' is contractible if the identity function, identity map 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 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 function, continuous Image (mathematics), image of some Separable space, separable metric space. ;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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
open cover has a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
collection of locally finite collections of subsets of ''X''. ;Cover (topology), 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: 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. ;Density: the minimal cardinality of a dense subset of a topological space. A set of density ℵ0 is a separable space. ;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 (topology), development. ;Development (topology), Development: A countable set, countable collection of open covers 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 sets, 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 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''  →  real number, 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: A space ''X'' is discrete space, discrete if every subset of ''X'' is open. We say that ''X'' carries the discrete topology.Steen & Seebach (1978) p.41 ;Discrete topology: See discrete space. ;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. ;Dunce hat (topology)

# E

;Entourage (topology), Entourage: See Uniform space. ;Exterior: The exterior of a set is the interior of its complement.

# F

;F-sigma set, ''F''σ set: An F-sigma set, ''F''σ set is a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
union of closed sets. ;Filter (mathematics), Filter: See also: Filters in topology. A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold: :# The empty set is not in ''F''. :# The intersection of any
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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: On a set ''X'' with respect to a family of functions into $X$, is the finest topology on ''X'' which makes those functions continuous function (topology), continuous. ;Fine topology (potential theory): On Euclidean space $\R^n$, the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. ;Finer topology: If ''X'' is a set, and if ''T''1 and ''T''2 are topologies on ''X'', then ''T''2 is finer topology, finer (or larger, stronger) than ''T''1 if ''T''2 contains ''T''1. Beware, some authors, especially mathematical analysis, analysts, use the term weaker. ;Finitely generated: See
Alexandrov topologyIn topology, an Alexandrov topology is a topological space, topology in which the intersection (set theory), intersection of any family of open sets is open. It is an axiom of topology that the intersection of any ''finite'' family of open sets is op ...
. ;First category: See Meagre set, Meagre. ;First-countable space, First-countable: A space is First-countable space, first-countable if every point has a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
local base. ;Fréchet: See T1. ;Frontier: See
Boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
. ;Full set: A compact space, compact subset ''K'' of the complex plane is called full if its complement (set theory), complement is connected. For example, the closed unit disk is full, while the unit circle is not. ;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X''  →  [0, 1] such that ''f''(''A'') = 0 and ''f''(''B'') = 1.

# G

;G-delta set, ''G''δ set: A G-delta set, ''G''δ set or inner limiting set is a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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: A generic point for a closed set is a point for which the closed set is the closure of the singleton set containing that point.

# H

; Hausdorff space, Hausdorff: A Hausdorff space (or T2 space, T2 space) is one in which every two distinct points have Disjoint sets, disjoint neighbourhoods. Every Hausdorff space is T1. ; H-closed space, 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 property, Hereditary: 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: If ''X'' and ''Y'' are spaces, a homeomorphism from ''X'' to ''Y'' is a bijection, bijective 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 space, Homogeneous: A space ''X'' is Homogeneous space, homogeneous 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 is homogeneous. ; homotopic, Homotopic maps: Two continuous maps ''f'', ''g'' : ''X''  →  ''Y'' are homotopic (in ''Y'') if there is a continuous map ''H'' : ''X'' × [0, 1]  →  ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' × [0, 1] is given the product topology. The function ''H'' is called a homotopy (in ''Y'') between ''f'' and ''g''. ; Homotopy: See homotopic, Homotopic maps. ; Hyperconnected space, 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. ; Quotient space (topology), Identification space: See Quotient space (topology), Quotient space. ; Indiscrete space: See Trivial topology. ; Infinite-dimensional topology: See Hilbert manifold 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 (topology), Interior: The interior (topology), interior 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 (topology), Interior. ; Isolated point: A point ''x'' is an isolated point 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 bijection, bijective 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 function, injective, although not every isometry is surjection, surjective.

# K

;Kolmogorov space, Kolmogorov axiom: See T0 space, T0. ;Kuratowski closure axioms: The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of ''X'' to its closure: :# ''Isotone function, Isotonicity'': Every set is contained in its closure. :# ''Idempotent function, Idempotence'': 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 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 point (mathematics), fixed points of this operator, i.e. a set ''A'' is closed if and only if ''c''(''A'') = ''A''. ;Kolmogorov topology :T''Kol'' = ∪; the pair (R,T''Kol'') is named ''Kolmogorov Straight''.

# L

;S and L spaces, L-space: An ''L-space'' is a Hereditary property#In topology, hereditarily Lindelöf space which is not hereditarily Separable space, separable. A Suslin line would be an L-space. ;Larger topology: See Finer topology. ;Limit point: A point ''x'' in a space ''X'' is a
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
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 space, Lindelöf: A space is Lindelöf space, Lindelöf if every open cover has a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
subcover. ;Local base: 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, metrisable, 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 space, Locally compact: A space is Locally compact space, locally compact 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: A space is locally connected if every point has a local base consisting of connected neighbourhoods. ; Locally dense: see ''Preopen''. ; Locally finite collection, Locally finite: A collection of subsets of a space is Locally finite collection, locally finite if every point has a neighbourhood which has nonempty intersection with only
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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: A space is locally path-connected if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected if and only if it is path-connected. ;Locally simply connected: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods. ;Loop (topology), Loop: If ''x'' is a point in a space ''X'', a loop (topology), 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 ''S''1 into ''X''.

# M

;Meagre set, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 space, Metacompact: A space is metacompact if every open cover has a point finite open refinement. ;Metric: See Metric space. ;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 metric map, strictly metric if the above inequality is strict for all ''x'' and ''y'' in ''X''. ;Metric space: A metric space (''M'', ''d'') is a set ''M'' equipped with a function ''d'' : ''M'' × ''M'' → real number, 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'') :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/Metrisable: A space is metrizable 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 (topology), Moore space: A Moore space (topology), Moore space is a developable space, developable regular Hausdorff space.Steen & Seebach (1978) p.163

# N

; Nearly open: see ''preopen''. ;Neighbourhood (mathematics), Neighbourhood/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.) ;Local base, Neighbourhood base/basis: See Local base. ;Neighbourhood system for a point ''x'': A neighbourhood system at a point ''x'' in a space is the collection of all neighbourhoods of ''x''. ;Net (mathematics), Net: A net (mathematics), net in a space ''X'' is a map from a directed set ''A'' to ''X''. A net from ''A'' to ''X'' is usually denoted (''x''α), where α is an index set, index variable ranging over ''A''. Every sequence is a net, taking ''A'' to be the directed set of natural numbers with the usual ordering. ;Normal space, Normal: A space is normal space, normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity. ;T4 space, Normal Hausdorff: A T4 space, normal Hausdorff space (or T4 space, T4 space) is a normal T1 space. (A normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff. ;Nowhere dense set, Nowhere dense: A nowhere dense set is a set whose closure has empty interior.

# O

; Open cover: An open cover 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 number, positive real number, 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: An open set is a member of the topology. ; Open map, Open function: A function from one space to another is open map, open if the image (mathematics), image of every open set is open. ; Open property: A property of points in a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is said to be "open" if those points which possess it form an open set. Such conditions often take a common form, and that form can be said to be an ''open condition''; for example, in metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".

# P

;Paracompact space, Paracompact: A space is paracompact space, paracompact 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: A partition of unity of a space ''X'' is a set of continuous functions from ''X'' to [0, 1] such that any point has a neighbourhood where all but a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. ;Path (topology), Path: A Path (topology), path in a space ''X'' is a continuous map ''f'' from the closed unit interval (mathematics), interval [0, 1] 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 space, Path-connected: A space ''X'' is path-connected space, path-connected 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 a set, partition of that space, which is partition of a set, finer than the partition into connected components. The set of path-connected components of a space ''X'' is denoted homotopy groups, 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 (topology), Closure. ;Polish space, Polish: 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 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. ;: 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: If $\left\left(X_i\right\right)$ is a collection of spaces and ''X'' is the (set-theoretic) Cartesian product of $\left\left(X_i\right\right),$ then the product topology 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: A proximity space (''X'', d) is a set ''X'' equipped with a binary relation 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 (''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 number, real-valued continuous function on the space is bounded. ;Pseudometric: See Pseudometric space. ;Pseudometric space: A pseudometric space (''M'', ''d'') is a set ''M'' equipped with a Real number, 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'', Set subtraction, minus . For instance, the interval (mathematics), interval (−1, 1) = is a neighbourhood of ''x'' = 0 in the real line, 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 space, compact. Some authors define "compact" to include the Hausdorff space, 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 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 ''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 doesn't imply that ''f'' is an open function. ;Quotient space (topology), Quotient space: If ''X'' is a space, ''Y'' is a set, and ''f'' : ''X'' → ''Y'' is any surjection, surjective function, then the Quotient topology 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 on ''X'', with ''Y'' the set of equivalence classes 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 (topology), refinement of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''. ; Regular space, Regular: A space is regular space, regular if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have Disjoint sets, disjoint neighbourhoods. ; T3 space, Regular Hausdorff: A space is T3 space, regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorff if and only if 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. ; Relatively compact: A subset ''Y'' of a space ''X'' is relatively compact 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is called resolvable space, resolvable if it is expressible as the union of two disjoint sets, disjoint dense subsets. ; Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.

# S

;S and L spaces, S-space: An ''S-space'' is a Hereditary property#In topology, hereditarily separable space which is not hereditarily Lindelöf space, Lindelöf. ;Scattered space, Scattered: A space ''X'' is scattered space, scattered if every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''. ;Scott continuity, Scott: The Scott topology on a
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is that in which the open sets are those Upper sets inaccessible by directed joins. ;Second category: See Meagre. ;Second-countable space, Second-countable: A space is second-countable space, second-countable or perfectly separable if it has a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
base for its topology. Every second-countable space is first-countable, separable, and Lindelöf. ;Semilocally simply connected: A space ''X'' is semilocally simply connected 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 space, Semiregular: A space is semiregular if the regular open sets form a base. ;Separable (topology), Separable: A space is separable (topology), separable if it has a
countable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
dense subset. ;Separated sets, Separated: Two sets ''A'' and ''B'' are separated sets, separated if each is Disjoint sets, disjoint from the other's closure. ;Sequentially compact: A space is sequentially compact if every sequence has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact. ;Short map: See metric map ;Simply connected space, Simply connected: A space is simply connected space, simply connected if it is path-connected and every loop is homotopic to a constant map. ;Smaller topology: See Coarser topology. ;Sober space, Sober: In a sober space, every hyperconnected space, irreducible closed subset is the closure (topology), closure of exactly one point: that is, has a unique generic point. ;Star: The star of a point in a given cover (topology), cover of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is the union of all the sets in the cover that contain the point. See star refinement. ;$f$-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. Beware, some authors, especially mathematical analysis, analysts, use the term weaker topology. ;Subbase: A collection of open sets is a subbase (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 Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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''. ;Subbase, Subbasis: See Subbase. ;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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an open set and a closed set. 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 space, irresolvable ;Subspace: If ''T'' is a topology on a space ''X'', and if ''A'' is a subset of ''X'', then the subspace topology 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 space, T0: A space is T0 space, 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 space, T1: A space is T1 space, 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 space, T2: See Hausdorff space. ;T3 space, T3: See T3 space, Regular Hausdorff. ;Tychonoff space, T: See Tychonoff space. ;T4 space, T4: See T4 space, Normal Hausdorff. ;T5 space, T5: See T5 space, Completely normal Hausdorff. ;Category of topological spaces, Top: See Category of topological spaces. ;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: 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 is the study of topologically invariant abstract algebra constructions on topological spaces. ;Topological space: A
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
(''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following axioms: :# 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 spaces. ;Topology: See Topological space. ;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
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: The trivial topology (or indiscrete topology) on a set ''X'' consists of precisely the empty set and the entire space ''X''. ;Tychonoff space, Tychonoff: A Tychonoff space (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 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 space, Ultrametric: A metric is an ultrametric if it satisfies the following stronger version of the triangle inequality: 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 spaces, a uniform isomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''−1 are uniformly continuous. 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: A uniform space is a set ''X'' equipped with a nonempty collection Φ of subsets of the Cartesian product ''X'' × ''X'' satisfying the following axioms: :# 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.

# W

; Weak topology: The weak topology 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 mathematical analysis, analysts, use the term stronger topology. ; Weakly countably compact: A space is weakly countably compact (or limit point compact) if every Infinity, infinite 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 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 order, well-ordered.) ; Well-connected: See Ultra-connected. (Some authors use this term strictly for ultra-connected compact spaces.)

# Z

;Zero-dimensional: A space is zero-dimensional if it has a base of clopen sets.Steen & Seebach (1978) p.33