nLab topological space

Idea

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.

Definitions

We present first the

• standard definition

and then a list of different

• equivalent definitions.

Finally we mention genuine

• variants of the notion.

Standard definition

Definition

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

1. finite intersections,

2. arbitrary unions.

Remark

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 $T$ be a topology on $X$’, then they mean ‘Let $X$ be equipped with the structure of a topological space, and let $T$ be the collection of open sets in this space’.

Remark

Since $X$ 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 $U$ of $X$ contains the empty set and is contained in $X$

$$\emptyset \subset U \subset X \,,$$

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

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 $X$.

Definition

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

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

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

Remark

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

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

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 $X$ with a frame of open sets (the standard definition, given above), called a topology on $X$.

• A set $X$ 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 $X$.

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

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

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

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

• A set $X$ together with, for each $x \in X$, a filter $N_x$ on $X$, i.e., a collection of inhabited subsets of $X$ closed under finite intersections and also upward-closed ($U \in N_x$ and $U \subseteq V$ together imply $V \subseteq N_x$). If $U \in N_x$, we call $U$ a neighborhood of $x$. The remaining conditions on these neighborhood systems are that $x \in U$ for every $U \in N_x$, and that for every $U \in N_x$, there exists $V \in N_x$ such that $V \subseteq U$ and $V$ is a neighborhood of each point it contains. In this formulation, a subset $U \subseteq X$ 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 $F$ on $X$ converges to a point $x \in X$ if $N_x \subseteq F$. The point then is to characterize properties of convergence abstractly.

• A relational beta-module; that is, a lax algebra of the monad $\beta$ of ultrafilters on the (1,2)-category Rel of sets and binary relations. More explicitly, this means a set $X$ together with a relation called “convergence” between ultrafilters and points satisfying certain axioms. This exhibits it as a special sort of generalized multicategory, and also as a special sort of pseudotopological space. However, the equivalence of this definition to the traditional definition of a topological space requires the ultrafilter principle to be true.

• A set with a convergence relation between nets or filters (not just ultrafilters) and points, or even between transfinite sequences and points, satisfying appropriate axioms.

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

• A (topological) base on a set $X$ is a collection $\mathcal{B}$ of subsets of $X$ whose union is all of $X$, and such that whenever $B, C \in \mathcal{B}$ and $x \in B \cap C$, there exists $D \in \mathcal{B}$ such that $D \subseteq B \cap C$ and $x \in D$.

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

• A set $X$ 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 $\mathcal{S}$. It consists of all possible unions of all possible finite intersections of members of $\mathcal{S}$. This is called the topology generated by the subbase.

Variations

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 $T_2$-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.

There are also unified topological spaces which contains notions of apartness and nearness in addition to the usual notion of neighbourhood in a topological space.

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 $T_0$-separation axiom forces the type to be an h-set.

 In dependent type theory

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

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

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

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

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

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

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

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

  for all $x:A$, where

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

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

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

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

  for all $x:A$, where

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

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

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 $(U, \mathrm{El}_U)$, where the elements of $U$ represent $U$-small types, and then quantify over $U$. In the case of topological spaces, instead of the open sets being closed under arbitrary unions, the open sets are only closed under all $U$-small unions $\bigcup_{i:\mathrm{El}_U(I)} B(i)$ for $I:U$.

Definition

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

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

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

by

$$x \in V \coloneqq \mathrm{El}_\Omega((i_O(V))(x))$$

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

Examples

Special cases

• finite topological space

• discrete topological space

• codiscrete topological space

• cofinite topology

• order topologyspecialization topology, Scott topology

• compact topological space

• paracompact topological space, sigma-compact topological space, Lindelöf topological space

• spectral topological space

• CW-complex

• manifold

…

Specific examples

• empty space

• point space

• Sierpinski space

• simplex

• circle

• Cantor space

• long line, line with two origins

• K-topology

• projective space, real projective space, complex projective space

• classifying space

…

Example

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

Related concepts

• separation axioms

• finer topology, coarser topology

• first countable topological space, second countable topological space, separable topological space, Hausdorff topological space, topological manifold

• $\sigma$-topological space

• unified topological space

• topological property

• locale, topos

• ionad

• proximity space, uniform space, syntopogenous space

• quotient topology, quotient topological space

• subspace topology, topological subspace

• product topological space

• effective topological space, equilogical space

• topological G-space

• filter space

• connected topological spaces, simply connected topological space

• nilpotent topological space

• fractal

• sheaf on a topological space

• pretopological space

References

Historical origins

The general idea of topology goes back to:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique 2 1 (1895) 1–123 [gallica:12148/bpt6k4337198/f7, Engl: pdf, pdf]

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

• Felix Hausdorff, Grundzüge der Mengenlehre, Leipzig: Veit (1914), Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [ISBN:978-0-8284-0061-9, ark:/13960/t2891gn8g]

The more general definition – dropping Hausdorff’s $T_2$-separation axiom and formulated in terms of closure operators that preserve finite unions – is due to:

• Kazimierz Kuratowski, Sur l’opération Ā de l’Analysis Situs, Fundamenta Mathematicae 3 (1922) 182–199 [doi:10.4064/fm-3-1-182-199]

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

Further textbook accounts:

• John Kelley, General topology, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [ISBN:978-0-387-90125-1]

• James Dugundji, Topology, Allyn and Bacon 1966 (pdf)

• James Munkres, Topology, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, pdf]

• Richard E. Hodel (ed.), Set-Theoretic Topology, Academic Press (1977) [doi:10.1016/C2013-0-11355-4]

• Klaus Jänich, Topology, Undergraduate Texts in Mathematics, Springer (1984, 1999) [ISBN:9780387908922, doi:10.1007/978-3-662-10574-0, Chapters 1-2: pdf]

• Ryszard Engelking, General Topology, Sigma series in pure mathematics 6, Heldermann 1989 (ISBN 388538-006-4)

• Steven Vickers, Topology via Logic, Cambridge University Press (1989) (toc pdf)

• Glen Bredon, Topology and Geometry, Graduate Texts in Mathematics 139, Springer (1993) [doi:10.1007/978-1-4757-6848-0, pdf]

• Terry Lawson, Topology: A Geometric Approach, Oxford University Press (2003) (pdf)

• Anatole Katok, Alexey Sossinsky, Introduction to Modern Topology and Geometry (2010) [toc pdf, pdf]

and leading over to homotopy theory:

• Ioan Mackenzie James, General Topology and Homotopy Theory, Springer 1984

  (doi:10.1007/978-1-4613-8283-6)

On counterexamples in topology:

• Lynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, Springer 1971 (doi:10.1007/978-1-4612-6290-9)

With emphasis on category theoretic aspects of general topology, notably on $T_n$-reflections:

• Horst Herrlich, George Strecker, Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 (pdf), pages 255-341 in: C. E. Aull, R Lowen (eds.), Handbook of the History of General Topology. Vol. 1, Kluwer 1997 (doi:10.1007/978-94-017-0468-7)

• Tai-Danae Bradley, Tyler Bryson, John Terilla, Topology – A categorical approach, MIT Press 2020 (ISBN:9780262539357, web version)

See also:

• Alan Hatcher, Algebraic Topology

• Neil Strickland, A Bestiary of Topological Objects [pdf, pdf]

and see further references at algebraic topology.

Lecture notes:

• Friedhelm Waldhausen, Topologie (pdf)

• Alex Kuronya, Introduction to topology, 2010 (pdf)

• Anatole Katok, Alexey Sossinsky, Introduction to modern topology and geometry (pdf)

• Urs Schreiber, Introduction to Topology, Bonn 2017

• Michael Müger, Topology for the working mathematician, Nijmegen 2018 (pdf)

Basic topology set up in intuitionistic mathematics is discussed in

• Franka Waaldijk, modern intuitionistic topology, 1996 (pdf)

See also:

• Topospaces, a Wiki with basic material on topology.