I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map


$$Top(T,P) \rightarrow Top (T,A) \times\_{Top(T,B)} Top (T,C)$$


is a bijection of sets. Now both sides are equipped with a topology, so I'm wondering whether the map is an homemorphism.


It is clear that the map continuous (because it is induced by continous maps), so the question boils down to asking whether the inverse map, which takes two compatible maps into $A$ and $B$ and builds the map into $P$ is continuous.


P.S.: I will gladly change the title of the question, if someone comes up with a better idea.



Is the defining bijection for a pullback of topological spaces a homeomorphism?
Yes. The functor $Top(T,-)$ preserves limits because it is a right adjoint. 


