Homotopy like a path in Hom(X,Y)

Question

If we have to topological spaces X,Y and the space of continuous function Hom(X,Y) between them, the homotopy between to elements of Hom(X,Y) -- $F: I\rightarrow Hom(X,Y)$, where F(t) = f_t, $t\in [0,1]$ and $f_t = F(x,t)$. How can I proof, that this interpretation of homotopy is true, iff $X$ -- hausdorff space and locally compact space. I think, that this is somehow related to Exponential law (in topology).

Answer 1

One direction of your implication is proved in detail in the following nLab article: https://ncatlab.org/nlab/show/compact-open+topology

There are several references linked there. The converse is not true: there are spaces $X$ which are not locally compact Hausdorff which satisfy this property. At least the "core compact" spaces do, and I think there are more examples than that.