I realize most people work in "convenient categories" where this is not an issue.
In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with the compact-open topology): $F(X\times Y,Z)\cong F(X,F(Y,Z))$ when $X$ is Hausdorff and $Y$ is locally compact Hausdorff. There is also supposed to be a homeomorphism in the based case with the same conditions on $X$ and $Y$: $F_{\ast}(X\wedge Y,Z)\cong F_{\ast}(X,F_{\ast}(Y,Z))$involving spaces of based maps and the smash product $\wedge$. This, for instance, is asserted on n-lab. I checked the references listed on this page and many other texts but have not found a proof of this "well-known fact."
It seems pretty clear if $X$ and $Y$ are compact Hausdorff (EDIT: in fact this is Theorem 6.2.38 of Maunder's Algebraic Topology) but can this really be proven in this generality?
Can anyone provide a reference for a proof?