Category of topological spaces

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.

N.B. Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms.

Like many categories, the category Top is a concrete category (also known as a construct), meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor

to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function.

The forgetful functor U has both a left adjoint

which equips a given set with the discrete topology and a right adjoint

...
Wikipedia