# Category of topological spaces with continuous maps

This article describes a category (in the mathematical sense) whose objects are topological spaces, and whose morphisms are continuous maps. In other words, it gives a category structure to the collection of all topological spaces.
View other category structures on topological spaces

## Definition

The category of topological spaces with continuous maps, often simply called the category of topological spaces, is defined as follows:

Aspect Name Definition/description
objects topological spaces A topological space is a set along with a collection of subsets, called open subsets, that contains the empty subset and the whole space, and is closed under taking arbitrary unions and finite intersections.
morphisms continuous maps between topological spaces A continuous map $f:X \to Y$ of topological spaces is a set map with the property that for every open subset $U$ of $Y$, $f^{-1}(U)$ is an open subset of $X$.
composition law for morphisms compose as set maps Not needed.

This is the default category structure on the collection of topological spaces. The category is sometimes denoted $\operatorname{Top}$.

## Constructs in this category

Construct Name in this category Definition/description
isomorphism homeomorphism A homeomorphism is a continuous bijection whose inverse is continuous.
monomorphism homeomorphism onto its image, which is endowed with the subspace topology
epimorphism quotient map equivalent (?)
categorical product product as sets, endowed with the product topology
categorical coproduct disjoint union
initial object one-point space
final object empty space

## Enhancements of this category

New category/category-like structure Objects Morphisms Other aspects
category of based topological spaces with based continuous maps (also called category of pointed topological spaces with pointed maps) based topological spaces: a based topological space is a topological space with a specially marked point in it called its basepoint based continuous maps: a continuous map of the topological spaces that sends the basepoint to the basepoint.
2-category of topological spaces with continuous maps and homotopies topological spaces continuous maps the 2-morphisms are homotopies

## Functors from this category

Target category Name/description of functor Behavior on objects Behavior on morphisms
category of sets with set maps forgetful functor from category of topological spaces with continuous maps to category of sets sends a topological space to its underlying set, i.e., forgets the topology. sends a continuous map to the same map, now viewed as a set map between the underlying sets.
homotopy category of topological spaces homotopy functor from category of topological spaces to homotopy category of topological spaces sends a topological space to itself sends a map of topological spaces to its homotopy class.
category of chain complexes with chain maps singular chain complex functor sends a topological space to its singular chain complex a map of topological spaces induces a chain map by first inducing a map on singular simplices via composition and then inducing a map on the chain groups which are freely generated by these simplices.

## Functors to this category

Target category Name/description of functor Behavior on objects Behavior on morphisms
category of based topological spaces with based continuous maps forget the basepoint sends a based topological space to the same topological space, but we no longer remember the basepoint. sends a based continuous map to the same continuous map.
category of pairs of topological space and subspace keep the bigger space sends a pair to the big space part of the pair. sends a continuous map of pairs to the continuous map at the level of the bigger space.
category of pairs of topological space and subspace keep the smaller space sends a pair to the small space part of the pair. sends a continuous map of pairs to the continuous map at the level of the smaller space.
category of cellular spaces with cellular maps
category of simplicial complexes with simplicial maps