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 category of compactly generated weak Hausdorff spaces.
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.
Topological categories have many properties in common with Top (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
In fact, the forgetful functor U : Top → Set uniquely lifts both limits and colimits and preserves them as well.
Specifically, if F is a diagram in Top and (L, φ : 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, φ : L → F).
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.