Category of topological spaces

From Academic Kids

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. 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.

Contents

Top is a concrete category

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

U : TopSet

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

Limits and colimits

The category Top is both complete and cocomplete, which means that all small limits and colimits exist in Top.

The forgetful functor U : TopSet has a left adjoint which equips a given set with the discrete topology and a right adjoint which equips a given set with the trivial topology. This implies that the functor U is both limit-preserving and colimit-preserving, i.e. limits in Top are given by placing topologies on the corresponding limits in Set.

Examples of limits and colimits in Top include:

Other properties

Relationships to other categories

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