Editing Category of topological spaces (section)

{{Short description|Category whose objects are topological spaces and whose morphisms are continuous maps}}
In [[mathematics]], the '''category of topological spaces''', often denoted '''Top''', is the [[category (category theory)|category]] whose [[object (category theory)|object]]s are [[topological space]]s and whose [[morphism]]s are [[continuous map]]s.  This is a category because the [[function composition|composition]] of two continuous maps is again continuous, and the identity function is continuous. The study of '''Top''' and of properties of [[topological space]]s using the techniques of [[category theory]] is known as '''categorical topology'''.

N.B. Some authors use the name '''Top''' for the categories with [[topological manifold]]s, with 
[[compactly generated space|compactly generated spaces]] as objects and continuous maps as morphisms or with the [[category of compactly generated weak Hausdorff spaces]].