nLab topological space

Topological spaces



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Topological spaces


The notion of topological space aims to axiomatize the idea of a space as a collection of points that hang together (“cohere”) in a continuous way. Roughly speaking, a topology on a set “of points” prescribes which subsets are to be considered “neighborhoods” of the points they contain. Various conditions or axioms must be satisfied in order for such neighborhood systems to form a topology, but one of the most important is that for any two neighborhoods of a point, their intersection must also be a neighborhood of that point.

Many notions of spaces used in mathematics have underlying topological spaces, such as: manifolds, schemes, probability spaces, etc.

The concept of a topology, gradually refined over the latter half of the 19th century and the first two decades of the 20th, was developed to capture what it means abstractly for a mapping between sets of points to be “continuous”. Intuitively, the idea of bending, twisting or crumpling a continuous body applies to continuous mappings, because they preserve neighborhood relations (in a suitable sense), but tearing, for instance, does not.

For example, the surface of a torus or doughnut is topologically equivalent to the surface of a mug: the surface of the mug can be deformed continuously into the surface of a torus. Abstractly speaking: the continuous cohesion among the collections of points of the two surfaces is the same. Similarly, a circle and a square are considered equivalent from the standpoint of their topologies.

Some one-dimensional shapes with different topologies: the Mercedes-Benz symbol, a line, a circle, a complete graph with 5 nodes, the skeleton of a cube, and an asterisk (or, if you’ll permit the one-dimensional approximation, a starfish). On the other hand, a circle has the same topology as a line segment with a wormhole at its finish which teleports you to its start; or more prosaically: The circle is homeomorphic to the closed interval with endpoints identified.

There is a generalization of the notion of topological spaces to that of locales, which consists of dropping the assumption that all neighbourhoods are explicitly or even necessarily supported by points. For this reason, the theory of locales is sometimes called “pointless topology”. In this form, the definition turns out to be quite fundamental and can be naturally motivated from just pure logic – as the formal dual of frames – as well as, and dually, from category theory in its variant as topos theory – by the notion of (0,1)-toposes.

Topological spaces are the objects studied in topology. But types of topological spaces exist in such great and wild profusion that in practice it is often more convenient to replace strict topological equivalence by a notion of weak equivalence, namely of weak homotopy equivalence. From this point of view, topological spaces support also homotopy theory.

Topological spaces equipped with extra properties and structure form the fundament of much of geometry. For instance a topological space locally isomorphic to a Cartesian space is a manifold. A topological space equipped with a notion of smooth functions into it is a diffeological space. The intersection of these two notions is that of a smooth manifold on which differential geometry is based. And so on.


We present first the

and then a list of different

Finally we mention genuine

Standard definition


A topological space is a set XX equipped with a set of subsets UXU \subset X, called the open sets, which are closed under

  1. finite intersections,

  2. arbitrary unions.


The word ‘topology’ sometimes means the study of topological spaces but here it means the collection of open sets in a topological space. In particular, if someone says ‘Let TT be a topology on XX’, then they mean ‘Let XX be equipped with the structure of a topological space, and let TT be the collection of open sets in this space’.


Since XX itself is the intersection of zero subsets, it is open, and since the empty set \emptyset is the union of zero subsets, it is also open. Moreover, every open subset UU of XX contains the empty set and is contained in XX

UX, \emptyset \subset U \subset X \,,

so that the topology of XX is determined by a poset of open subsets Op(X)Op(X) with bottom element =\bot = \emptyset and top element =X\top = X.

Since by definition the elements in this poset are closed under finite meets (intersection) and arbitrary joins (unions), this poset of open subsets defining a topology is a frame, the frame of opens of XX.


A homomorphism between topological spaces f:XYf : X \to Y is a continuous function:

a function f:XYf:X\to Y of the underlying sets such that the preimage of every open set of YY is an open set of XX.

Topological spaces with continuous maps between them form a category, usually denoted Top.


The definition of continuous function f:XYf : X \to Y is such that it induces a homomorphism of the corresponding frames of opens the other way around

Op(X)Op(Y):f 1. Op(X) \leftarrow Op(Y) : f^{-1} \,.

And this is not just a morphism of posets but even of frames. For more on this see at locale.

Alternate equivalent definitions

There are many equivalent ways to define a topological space. A non-exhaustive list follows:

  • A set XX with a frame of open sets (the standard definition, given above), called a topology on XX.

  • A set XX with a co-frame of closed sets (the complements of the open sets), satisfying dual axioms: closure under finite unions and arbitrary intersections. This is sometimes called a co-topology on XX.

  • A pair (X,int)(X, int), where int:P(X)P(X)int\colon P(X) \to P(X) is a left exact comonad on the power set of XX (the “interior operator”). In more nuts-and-bolts terms, this means for all subsets A,BA, B of XX we have

    ABint(A)int(B),int(A)A,int(A)int(int(A)),int(AB)=int(A)int(B),int(X)=X.A \subseteq B \Rightarrow int(A) \subseteq int(B), \;\;\; int(A) \subseteq A, \;\;\; int(A) \subseteq int(int(A)), \;\;\; int(A \cap B) = int(A) \cap int(B), \;\;\; int(X) = X.

    The open sets are exactly the fixed points of intint. The first three of these conditions say intint is a coclosure operator.

  • A pair (X,cl)(X, cl) where clcl is a right exact Moore closure operator satisfying axioms dual to those of intint. The closed sets are the fixed points of clcl. Such an operator is sometimes called a Kuratowski closure operator (compare Kuratowski’s closure-complement problem at closed subspace).

  • A set XX together with, for each xXx \in X, a filter N xN_x on XX, i.e., a collection of inhabited subsets of XX closed under finite intersections and also upward-closed (UN xU \in N_x and UVU \subseteq V together imply VN xV \subseteq N_x). If UN xU \in N_x, we call UU a neighborhood of xx. The remaining conditions on these neighborhood systems are that xUx \in U for every UN xU \in N_x, and that for every UN xU \in N_x, there exists VN xV \in N_x such that VUV \subseteq U and VV is a neighborhood of each point it contains. In this formulation, a subset UXU \subseteq X is open if it is a neighborhood of every point it contains.

The next two definitions of topological space are at a higher level of abstraction, but the underlying idea that connects them with the neighborhood system formulation is that we say a filter FF on XX converges to a point xXx \in X if N xFN_x \subseteq F. The point then is to characterize properties of convergence abstractly.

The following are not definitions, but they provide alternative ways to present a topological space.

  • A (topological) base on a set XX is a collection \mathcal{B} of subsets of XX whose union is all of XX, and such that whenever B,CB, C \in \mathcal{B} and xBCx \in B \cap C, there exists DD \in \mathcal{B} such that DBCD \subseteq B \cap C and xDx \in D.

If \mathcal{B} is a base on XX, then it is easily shown that the collection of all unions of subcollections of \mathcal{B} is a topology on XX.

  • A set XX with any collection of subsets whatsoever, to be thought of as a subbase for a topology.

From the fact that the intersection of any collection of topologies is also a topology, there is a smallest topology that contains a given subbase 𝒮\mathcal{S}. It consists of all possible unions of all possible finite intersections of members of 𝒮\mathcal{S}. This is called the topology generated by the subbase.


Historically, the notion of topological space (see the historical references given there) involving neighbourhoods was first developed by Felix Hausdorff in 1914 in his seminal text on set theory and topology, Fundamentals of Set Theory (Grundzüge der Mengenlehre). Hausdorff’s definition originally contained the T 2T_2-separation axiom (now known as the definition of Hausdorff spaces). This axiom was in effect removed by Kazimierz Kuratowski in 1922, who defined general topological spaces in terms of closure operators that preserve finite unions. The usual open set formulation was widely popularized by Bourbaki in their 1940 treatise (without identifying a single author behind this notion).

However, in more modern treatments that emphasize category theoretic methods, particularly to address needs of homotopy theory, it becomes important to consider not just the category Top of all topological spaces, but convenient categories of topological spaces that are better behaved, especially with regard to function spaces and cartesian closure. Thus many texts work with nice topological spaces (such as sequential topological spaces) and/or a nice- or convenient category of topological spaces (such as compactly generated spaces), or indeed to directly use a model of \infty-groupoids (such as simplicial sets).

On the other hand, when doing topos theory or working in constructive mathematics, it is often more appropriate to use locales than topological spaces.

Some applications to analysis require more general convergence spaces or other generalisations.

In dependent type theory, one could also have a topological space be a general type instead of an h-set. For most kinds of topological spaces in dependent type theory, the T 0 T_0 -separation axiom forces the type to be an h-set.

 In dependent type theory

In dependent type theory, given a type XX, the type of all subtypes of XX, the powerset of XX, is defined as the function type

𝒫(X)XΩ\mathcal{P}(X) \coloneqq X \to \Omega

where Ω\Omega is the type of all propositions with the type reflector type family P:ΩEl Ω(P)typeP:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}. In the inference rules for the type of all propositions, one has an operation () Ω(-)_\Omega which takes a proposition PP and turns it into an element of the type of all propositions P Ω:ΩP_\Omega:\Omega.

The local membership relation x ABx \in_A B between elements x:Ax:A and material subtypes B:𝒫(A)B:\mathcal{P}(A) is defined as

x ABEl Ω(B(x))x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))

Arbitrary unions and intersections of subtypes could be defined in dependent type theory:

  • Given a type AA and a type II, there is an function :(I𝒫(A))𝒫(A)\bigcap:(I \to \mathcal{P}(A)) \to \mathcal{P}(A) called the II-indexed intersection, such that for all families of subtypes B:I𝒫(A)B:I \to \mathcal{P}(A), i:IB(i)\bigcap_{i:I} B(i) is defined as

    ( i:IB(i))(x)(i:I.x AB(i)) Ω\left(\bigcap_{i:I} B(i)\right)(x) \coloneqq (\forall i:I.x \in_A B(i))_\Omega

    for all x:Ax:A, where

    x:A.B(x)[ x:AB(x)]\forall x:A.B(x) \coloneqq \left[\prod_{x:A} B(x)\right]

    is the universal quantification of a type family and [T][T] is the propositional truncation of TT.

  • Given a type AA and a type II, there is an function :(I𝒫(A))𝒫(A)\bigcup:(I \to \mathcal{P}(A)) \to \mathcal{P}(A) called the II-indexed union, such that for all families of subtypes B:I𝒫(A)B:I \to \mathcal{P}(A), i:IB(i)\bigcup_{i:I} B(i) is defined as

    ( i:IB(i))(x)(i:I.x AB(i)) Ω\left(\bigcup_{i:I} B(i)\right)(x) \coloneqq (\exists i:I.x \in_A B(i))_\Omega

    for all x:Ax:A, where

    x:A.B(x)[ x:AB(x)]\exists x:A.B(x) \coloneqq \left[\sum_{x:A} B(x)\right]

    is the existential quantification of a type family and [T][T] is the propositional truncation of TT.

In dependent type theory, however, one cannot quantify over arbitrary types, since one could only quantify over elements of a type. Instead, one has to use a Tarski universe (U,El U)(U, \mathrm{El}_U), where the elements of UU represent UU-small types, and then quantify over UU. In the case of topological spaces, instead of the open sets being closed under arbitrary unions, the open sets are only closed under all UU-small unions i:El U(I)B(i)\bigcup_{i:\mathrm{El}_U(I)} B(i) for I:UI:U.


Given a Tarski universe (U,El U)(U, \mathrm{El}_U), a topological space is a type XX with a UU-small topology, a type of subtypes O(X)O(X) with canonical embedding i O:O(X)𝒫(X)i_O:O(X) \hookrightarrow \mathcal{P}(X), called the open sets of XX, which are closed under finite intersections and UU-small unions.

Given a topological space (X,O(X))(X, O(X)), we define the membership relation between elements x:Xx:X and open sets V:O(X)V:O(X):

x:X,V:O(X)xVtypex:X, V:O(X) \vdash x \in V \; \mathrm{type}


xVEl Ω((i O(V))(x))x \in V \coloneqq \mathrm{El}_\Omega((i_O(V))(x))

By definition of the type of all propositions and its type reflector, xVx \in V is always a h-proposition for all x:Xx:X and V:O(X)V:O(X).


Special cases

Specific examples


The Cartesian space n\mathbb{R}^n with its standard notion of open subsets generated from: unions of open balls D n nD^n \subset \mathbb{R}^n.


Historical origins

The general idea of topology goes back to:

The notion of topological space involving neighbourhoods was first developed, for the special case now known as Hausdorff spaces, in:

The more general definition – dropping Hausdorff’s T 2T_2-separation axiom and formulated in terms of closure operators that preserve finite unions – is due to:

The modern formulation via open set was widely popularized by:

  • Nicolas Bourbaki], Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques. Actualités scientifiques et industrielles, vol. 858. Hermann, Paris (1940)

    General topology, Elements of Mathematics III, Springer (1971, 1990, 1995) [doi:10.1007/978-3-642-61701-0]


Further textbook accounts:

and leading over to homotopy theory:

On counterexamples in topology:

With emphasis on category theoretic aspects of general topology, notably on T nT_n-reflections:

See also:

and see further references at algebraic topology.

Lecture notes:

Basic topology set up in intuitionistic mathematics is discussed in

See also:

  • Topospaces, a Wiki with basic material on topology.

Last revised on June 24, 2024 at 10:12:02. See the history of this page for a list of all contributions to it.