# 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. 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 categories with topological manifolds or with compactly generated spaces as objects and continuous maps as morphisms.

Like many categories, the category **Top** is a concrete category, 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.

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

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 between indiscrete spaces is continuous, both of these functors give full embeddings of **Set** into **Top**.

**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 category **Top** is both complete and cocomplete, which means that all small limits and colimits exist in **Top**. In fact, the forgetful functor *U* : **Top** → **Set** uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in **Top** are given by placing topologies on the corresponding (co)limits in **Set**.

Specifically, if *F* is a diagram in **Top** and (*L*, *φ* : *L* → *F*) is a limit of *UF* in **Set**, the corresponding limit of *F* in **Top** is obtained by placing the initial topology on (*L*, *φ* : *L* → *F*). Dually, colimits in **Top** are obtained by placing the final topology on the corresponding colimits in **Set**.

Unlike many *algebraic* categories, the forgetful functor *U* : **Top** → **Set** does not create or reflect limits since there will typically be non-universal cones in **Top** covering universal cones in **Set**.