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The following is a list of named topologies or
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, many of which are counterexamples in
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
and related branches of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. This is not a list of
properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an ...
that a topology or topological space might possess; for that, see List of general topology topics and
Topological property 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 space ...
.


Discrete and indiscrete

*
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 to ...
− All subsets are open. *
Indiscrete 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 conseque ...
, chaotic topology, or
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 conseque ...
− Only the empty set and its complement are open.


Cardinality and ordinals

*
Cocountable topology The cocountable topology, also known as the countable complement topology, is a topology that can be defined on any infinite set X. In this topology, a set is open if its complement in X is either countable or equal to the entire set. Equivalen ...
** Given a topological space (X, \tau), the '' '' on X is the topology having as a subbasis the union of and the family of all subsets of X whose complements in X are countable. * Cofinite topology * Double-pointed cofinite topology * Ordinal number topology *
Pseudo-arc In general topology, the pseudo-arc is the simplest nondegenerate hereditarily indecomposable continuum. The pseudo-arc is an arc-like homogeneous continuum, and played a central role in the classification of homogeneous planar continua. R. H. ...
* Ran space * Tychonoff plank


Finite spaces

* Discrete two-point space − The simplest example of a
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
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 ...
. *
Finite topological space In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are ...
* Pseudocircle − A
finite topological space In mathematics, a finite topological space is a topological space for which the underlying set (mathematics), point set is finite set, finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are ...
on 4 elements that fails to satisfy any
separation axiom In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes ...
besides T0. However, from the viewpoint of
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 ...
, it has the remarkable property that it is indistinguishable from the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
S^1. *
Sierpiński space In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The ...
, also called the connected two-point set − A 2-point set \ with the particular point topology \.


Integers

* Arens–Fort space − A Hausdorff, regular, normal space that is not first-countable or compact. It has an element (i.e. p := (0, 0)) for which there is no sequence in X \setminus \ that converges to p but there is a sequence x_\bull = \left(x_i\right)_^\infty in X \setminus \ such that (0, 0) is a cluster point of x_\bull. * Arithmetic progression topologies * The Baire space\N^ with the product topology, where \N denotes the
natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
endowed with the discrete topology. It is the space of all sequences of natural numbers. * Divisor topology *
Partition topology In mathematics, a partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names: * The is t ...
** Deleted integer topology ** Odd–even topology


Fractals and Cantor set

* Apollonian gasket *
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...
− A subset of the closed interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> with remarkable properties. ** Cantor dust **
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
*
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...
*
Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sie ...
* Mosely snowflake * Sierpiński carpet *
Sierpiński triangle The Sierpiński triangle, also called the Sierpiński gasket or Sierpiński sieve, is a fractal with the overall shape of an equilateral triangle, subdivided recursion, recursively into smaller equilateral triangles. Originally constructed as a ...
*
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
, also called the − A closed nowhere dense (and thus meagre) subset of the unit interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> that has positive
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
and is not a Jordan measurable set. The complement of the fat Cantor set in Jordan measure is a bounded open set that is not Jordan measurable.


Orders

*
Alexandrov topology In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...
* Lexicographic order topology on the unit square *
Order topology In mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, ...
** Lawson topology ** Poset topology ** Upper topology ** Scott topology ***
Scott continuity In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed su ...
* Priestley space * Roy's lattice space * Split interval, also called the ' and the ' − All compact separable ordered spaces are order-isomorphic to a subset of the split interval. It is compact Hausdorff, hereditarily Lindelöf, and hereditarily separable but not
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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
. Its metrizable subspaces are all countable. *
Specialization (pre)order In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the ...


Manifolds and complexes

* Branching line − A non-Hausdorff manifold. * Double origin topology * E8 manifold − A
topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...
that does not admit a
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold. Definition A smooth structure on a manifold M ...
. *
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative function \, \cdot ...
− The natural topology on
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\Reals^n induced by the
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is oc ...
, which is itself induced by the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
. **
Real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
\Reals **
Unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> *
Extended real number line In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential ...
* Fake 4-ball − A compact
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 t ...
topological 4-manifold. *
House with two rooms House with two rooms or Bing's house is a particular contractible, 2-dimensional simplicial complex that is not collapse (topology), collapsible. The name was given by R. H. Bing.Bing, R. H., ''Some Aspects of the Topology of 3-Manifolds Related ...
− A
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 t ...
, 2-dimensional
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
that is not collapsible. *
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
* Lens space * Line with two origins, also called the ' − It is a non-Hausdorff manifold. It is locally homeomorphic to
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
and thus locally metrizable (but not
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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
) and locally Hausdorff (but not Hausdorff). It is also a T1 locally regular space but not a semiregular space. * Prüfer manifold − A Hausdorff 2-dimensional real analytic manifold that is not
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
. *
Real projective line In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
*
Torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
** 3-torus ** Solid torus *
Unknot In the knot theory, mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a Knot (mathematics), knot tied into it, unknotted. To a knot ...
* Whitehead manifold − An open
3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
that is
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 t ...
, but not
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to \Reals^3.


Hyperbolic geometry

* Gieseking manifold − A cusped hyperbolic 3-manifold of finite volume. * Horosphere ** Horocycle * Picard horn *
Seifert–Weber space In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discov ...


Paradoxical spaces

* Lakes of Wada − Three disjoint connected open sets of \Reals^2 or (0, 1)^2 that all have the same boundary.


Unique

*
Hantzsche–Wendt manifold The Hantzsche–Wendt manifold, also known as the HW manifold or didicosm, is a compact space, compact, orientable, flat manifold, flat 3-manifold, first studied by Walter Hantzsche and Hilmar Wendt in 1934. It is the only closed flat 3-manifold wi ...
− A compact, orientable, flat 3-manifold. It is the only closed flat 3-manifold with first
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
zero.


Related or similar to manifolds

* Dogbone space * Dunce hat (topology) * Hawaiian earring *
Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain sense "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither ...
* Rose (topology)


Embeddings and maps between spaces

* Alexander horned sphere − A particular embedding of a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
into 3-dimensional Euclidean space. * Antoine's necklace − A topological embedding of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...
in 3-dimensional Euclidean space, whose complement is not
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 (topology), path between two points can be continuously transformed into any other such path while preserving ...
. * Irrational winding of a torus/ Irrational cable on a torus *
Knot (mathematics) In mathematics, a knot is an embedding of the circle () into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation o ...
* Linear flow on the torus *
Space-filling curve In mathematical analysis, a space-filling curve is a curve whose Range of a function, range reaches every point in a higher dimensional region, typically the unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Pea ...
*
Torus knot In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...
*
Wild knot In the mathematical theory of knots, a knot is tame if it can be "thickened", that is, if there exists an extension to an embedding of the solid torus S^1\times D^2 into the 3-sphere. A knot is tame if and only if it can be represented as a fin ...


Counter-examples (general topology)

The following topologies are a known source of counterexamples for
point-set topology In mathematics, general topology (or point set 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 differ ...
. * Alexandroff plank * Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither first countable,
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 e ...
, nor countably compact. * Arens square * Bullet-riddled square - The space
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
2 \setminus \Q^2, where
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
2 \cap \Q^2 is the set of bullets. Neither of these sets is Jordan measurable although both are Lebesgue measurable. * Cantor tree * Comb space * Dieudonné plank * Double origin topology * Dunce hat (topology) * Either–or topology *
Excluded point topology In mathematics, the excluded point topology is a topological space, topology where exclusion of a particular point defines open set, openness. Formally, let ''X'' be any non-empty set and ''p'' ∈ ''X''. The collection :T = \ \cup \ of subsets of ...
− A topological space where the open sets are defined in terms of the exclusion of a particular point. *
Fort space In mathematics, there are a few topological spaces named after M. K. Fort, Jr. Fort space Fort space is defined by taking an infinite set ''X'', with a particular point ''p'' in ''X'', and declaring open the subsets ''A'' of ''X'' such that: * ...
* Half-disk topology *
Hilbert cube In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ca ...
, 1/1\times , 1/2\times , 1/3\times \cdots with 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-seemin ...
. * Infinite broom * Integer broom topology * K-topology * Knaster–Kuratowski fan *
Long line (topology) In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain sense "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither ...
* Moore plane, also called the ' − A first countable, separable, completely regular, Hausdorff, Moore space that is not normal, Lindelöf,
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, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
, second countable, nor
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 e ...
. It also an uncountable closed subspace with the discrete topology. * Nested interval topology * Overlapping interval topology − Second countable space that is T0 but not T1. * Particular point topology − Assuming the set is infinite, then contains a non-closed compact subset whose closure is not compact and moreover, it is neither metacompact nor
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
. * Rational sequence topology * Sorgenfrey line, which is \Reals endowed with
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
− It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, Baire, and a Moore space but not metrizable, second-countable, σ-compact, nor locally compact. * Sorgenfrey plane, which is the product of two copies of the Sorgenfrey line − A Moore space that is neither normal,
paracompact In mathematics, a paracompact space is a topological space in which every open cover has an open Cover (topology)#Refinement, refinement that is locally finite collection, locally finite. These spaces were introduced by . Every compact space is par ...
, nor second countable. *
Topologist's sine curve In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the functi ...
* Tychonoff plank * Vague topology *
Warsaw circle Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory ...


Topologies defined in terms of other topologies


Natural topologies

List of natural topologies. * Adjunction space * Disjoint union (topology) * Extension topology *
Initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that ...
*
Final topology In general topology and related areas of mathematics, the final topology (or coinduced, weak, colimit, or inductive topology) on a Set (mathematics), set X, with respect to a family of functions from Topological space, topological spaces into X, is ...
*
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-seemin ...
*
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 to ...
*
Subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
* Weak topology


Compactifications

Compactifications include: *
Alexandroff extension In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
**
Projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standard ...
* Bohr compactification * Eells–Kuiper manifold *
Projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standard ...
*
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Ha ...
** Stone topology ** Stone–Čech remainder * Wallman compactification


Topologies of uniform convergence

This lists named topologies of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
. *
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 ...
** Loop space * Interlocking interval topology * Modes of convergence ( annotated index) * Operator topologies *
Pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
** Weak convergence (Hilbert space) ** Weak* topology * Polar topology * Strong dual topology *
Topologies on spaces of linear maps In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The art ...


Other induced topologies

* Box topology * Compact complement topology * Duplication of a point: Let x \in X be a non-
isolated point In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton i ...
of X, let d \not\in X be arbitrary, and let Y = X \cup \. Then \tau = \ is a topology on Y and x and d have the same
neighborhood filter 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 ...
s in Y. In this way, x has been duplicated. * Extension topology


Functional analysis

* Auxiliary normed spaces * Finest locally convex topology * Finest vector topology * Helly space *
Mackey topology In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not ...
* Polar topology * Vague topology


Operator topologies

* Dual topology * Norm topology * Operator topologies *
Pointwise convergence In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
** Weak convergence (Hilbert space) ** Weak* topology * Polar topology * Strong dual space * Strong operator topology *
Topologies on spaces of linear maps In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves. The art ...
* Ultrastrong topology * Ultraweak topology/ weak-* operator topology * Weak operator topology


Tensor products

* Inductive tensor product * Injective tensor product * Projective tensor product * Tensor product of Hilbert spaces * Topological tensor product


Probability

* Émery topology


Other topologies

* Erdős space − A Hausdorff,
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
, one-dimensional topological space X that is homeomorphic to X \times X. * Half-disk topology * Hedgehog space *
Partition topology In mathematics, a partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names: * The is t ...
*
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...


See also

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Citations


References

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External links


π-Base: An Interactive Encyclopedia of Topological Spaces
{{DEFAULTSORT:Topologies General topology Mathematics-related lists Topological spaces