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Encyclopedia > Category of topological spaces

The category Top has topological spaces as objects and continuous maps as morphisms. 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.

(Some authors use the name Top for the category with topological manifolds as objects and continuous maps as morphisms; Wikipedia follows the convention given above.)

The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms.

The empty set (considered as a topological space) is the initial object of Top; any singleton topological space is a terminal object. There are thus no zero objects in Top. Indeed, there aren't even zero morphisms in Top, and in particular the category is not preadditive.

The product in Top is given by the product topology on the cartesian product. Using the subspace topology for subsets of those products, one can then show that Top is a complete category. The coproduct is given by the disjoint union of topological spaces. By using the quotient topology, one can then show that Top is also cocomplete.

Top is not cartesian closed (and therefore also not a topos) since it does not have exponential objects.

We have a "forgetful" functor TopSet which assigns to each topological space the underlying set, and to each continuous map the underlying function. This functor is faithful, and therefore Top is a concrete category. The forgetful functor 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). Results from FactBites:

 Topological space - Wikipedia, the free encyclopedia (1777 words) Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. The category of topological spaces, Top, with topological spaces as objects and continuous functions as morphisms is one of the fundamental categories in mathematics. Sierpiński space is the simplest non-trivial, non-discrete topological space.
 Category of topological spaces - Wikipedia, the free encyclopedia (535 words) In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. The coproduct is given by the disjoint union of topological spaces.
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