Category of based topological spaces with based continuous maps

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

Definition

The category of based topological spaces with based continuous maps, also called the category of based topological spaces, the category of pointed topological spaces with pointed continuous maps, or the category of pointed topological spaces, is defined as follows:

Aspect	Name	Definition/description
objects	based topological spaces (also called pointed spaces)	A based topological space $(X,x_{0})$ is the following data: a topological space] $X$ along with a chosen point $x_{0}\in X$, called the basepoint.
morphisms	based continuous maps between based topological spaces	A continuous map of the topological spaces that sends the basepoint to the basepoint.
composition law for morphisms	compose as set maps	Not needed.

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