As already hinted at above, one can go even further. The category Top of topological spaces has as morphisms the continuous functions, where a function "f" is continuous if the inverse image "f" −1("O") of any open set in the codomain of "f" is open in the domain of "f". Thus any continuous function "f" from a space "X" to a space "Y" defines an inverse mapping "f" −1 from Ω("Y") to Ω("X"). Furthermore, it is easy to check that "f" −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a "morphism of frames". If we define Ω("f") = "f" −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from Frm to Top which is adjoint to Ω.