Category of topological spaces

In mathematics, the category of topological spaces, often denoted ${\displaystyle \mathbf {Top} }$, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous.

N.B. Some authors use the name ${\displaystyle \mathbf {Top} }$ for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces.