Topological spaces

Definition

A topological space is a set $X$ equipped with a set of subsets of $X$, called open sets, which are closed under finite intersections and arbitrary unions. Since $X$ itself is the intersection of zero subsets, it is open, and since the empty set $\varnothing$ is the union of zero subsets, it is also open.

The word ‘topology’ sometimes means the study of topological spaces but sometimes 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’.

The morphisms between topological spaces are continuous maps: functions $f:X\to Y$ such that the preimage of any open set is open.

Alternate 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 $(X,\mathrm{int})$, where $\mathrm{int}:P(X)\to P(X)$ is a left exact comonad on the power set of $X$ (the “interior operator”). The open sets are exactly the fixed points of $\mathrm{int}$.

• A pair $(X,\mathrm{cl})$ where $\mathrm{cl}$ is a right exact Moore closure operator satisfying axioms dual to those of $\mathrm{int}$. The closed sets are the fixed points of $\mathrm{cl}$.

• A relational $\beta$-module; that is, a lax algebra? of the monad 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.

• 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 and/or a nice category of spaces, or indeed to directly use a model of $\mathrm{\infty }$-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

• manifold

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Specific examples

• Cantor space

• long line

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