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.
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, limits in Top are given by placing topologies on the corresponding limits in Set. Specifically, if F is a diagram in Top and is a limit of UF in Set, the corresponding limit of F in Top is obtained by placing the initial topology on. 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. Examples of limits and colimits in Top include:
Direct limits and inverse limits are the set-theoretic limits with the final topology and initial topology respectively.
Adjunction spaces are an example of pushouts in Top.
Other properties
The monomorphisms in Top are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms.
The extremal monomorphisms are the subspace embeddings. In fact, in Top all extremal monomorphisms happen to satisfy the stronger property of being regular.
The extremal epimorphisms are the quotient maps. Every extremal epimorphism is regular.
The split monomorphisms are the inclusions of retracts into their ambient space.
The split epimorphisms are the continuous surjective maps of a space onto one of its retracts.
Top contains the important category Haus of Hausdorff spaces as a full subcategory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with denseimages in their codomains, so that epimorphisms need not be surjective.
Top contains the full subcategory CGHaus of compactly generated Hausdorff spaces, which has the important property of being a Cartesian closed category while still containing all of the typical spaces of interest. This makes CGHaus a particularly convenient category of topological spaces that is often used in place of Top.
The forgetful functor to Set has both a left and a right adjoint, as described above in the concrete category section.
There is a functor to the category of localesLoc sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of sober spaces and spatial locales.
