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.

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). A circle has the same topology as a line segment with a wormhole at its finish which teleports you to its start. You can see this by putting overlapping open intervals on each of the shapes. You’ll see that they respond the same way, so they’re equivalent in that sense.

The surface of a torus is also topologically equivalent to the surface of a mug. You can see this by putting open circles or slightly looser loops all over each surface: you’ll see that they also respond in the same way. Abstractly, the surface of the mug can be deformed continuously to become the standard torus: the continuous cohesion among the collections of points of the two surfaces is the same.

There is a slight generalization of the notion of topological space to that of a locale, which consists of dropping the assumption that all neighbourhoods are explicitly or even necessarily supported by points. In this form the definition is 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. By equipping them with a notion of weak equivalence, namely of weak homotopy equivalence, they turn out to support also homotopy theory.

Topological spaces equipped with extra property 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 (as above).

• A set $X$ with a co-frame of closed sets (the complements of the open sets) satisfying dual axioms.

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

• 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”). The open sets are exactly the fixed points of $int$.

• 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 $cl$.

• 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.

• 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.

Variations

The definition of topological space was a matter of some debate, especially about 100 years ago. Our definition is due to Bourbaki, so may be called Bourbaki spaces.

For some purposes, including homotopy theory, it is important to use 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.

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

• topological property

• locale, topos

• 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

References

See at topology.