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 $\operatorname {Top}$.

Constructions 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 (?)

Enhancements of this category

New category/category-like structure	Objects	Morphisms	Other aspects
category of based topological spaces with based 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.