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Concept
Category of topological spaces
Summary
In mathematics, the category of topological spaces, often denoted Top, is the whose s 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 is known as categorical topology.
N.B. Some authors use the name Top for the categories with topological manifolds, with
compactly generated spaces as objects and continuous maps as morphisms or with the .
Like many categories, the category Top is a , 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 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 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.
Top is the model of what is called a . 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 properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
Official source
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