View source for Category of topological spaces

The '''category of topological spaces''', denoted <math>\mathsf{Top}</math> or <math>\mathbf{Top}</math>, is the [[category]] whose objects are [[Topological space|topological spaces]] and whose [[Morphism|morphisms]] are [[Continuous function|continuous functions]].

== Definition ==
The category <math>\mathsf{Top}</math> consists of:

* <math>\operatorname{ob}(\mathsf{Top})</math> consists of all topological spaces,
* <math>\operatorname{mor}(\mathsf{Top})</math> consists of all continuos functions.

== Subcategories ==
Several important subcategories of <math>\mathsf{Top}</math> arise by restricting objects:

* <math>\mathsf{Haus}</math>, [[category of Hausdorff spaces]];
* <math>\mathsf{Comp}</math>, [[category of Compact spaces]];

* <math>\mathsf{Top}_{*}</math>, [[category of pointed topological spaces]];
* <math>\mathsf{CW}</math>, [[category of CW complexes]] or cell complexes.