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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 the model of what is called a topological category. 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).
Limits and colimits
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, ? : 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, ? : 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.
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.