 They can be used as a foundation for higher category theory, where they can play the role of (∞,1)-categories.An important example of a topological category in this sense is given by the category of CW complexes, where each set Hom("X","Y") of continuous maps from "X" to "Y" is equipped with the compact-open topology.In another approach, a topological category is defined as a category formula_1 along with a forgetful functor formula_2 that maps to the category of sets and has the following three properties