Category of Topological Spaces - As A Concrete Category

As A Concrete Category

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

U : Top → Set

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

D : Set → Top

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

I : Set → Top

which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top.

The construct Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on X while the least element is the indiscrete topology.

The construct Top is the model of what is called a topological category. These categories are characterized by the fact that every structured source has a unique initial lift . In Top the initial lift is obtained by placing the initial topology on the source. Topological categories have many nice properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).