$\mathrm{DISCLAIMER~:~}$I am not interested in working with compactly generated spaces.
This post is related to this one : Exponential Law for based spaces. I learned about the exponential law for topological spaces quite some time ago, and I thought I understood it well until I decided to reprove it today as I have been using it lately.
What confuses me is that I 'seem' to have proven it with weaker conditions than those stated in the textbooks. To be precise, I think I have shown that there is a natural homeomorphism $\mathrm{Map}(X\times Y,Z)\simeq\mathrm{Map}(X,\mathrm{Map}(Y,Z))$ where $Z$ is any topological space, $X$ is Hausdorff and $Y$ locally compact $no$ $Hausdorff$ $condition$ $required\dots$ In all textbooks I 'm familiar with, none of which feature a proof of the above fact, the extra assumption is made that $Y$ be Hausdorff. The proof I gave is, I think, the one I learnt in Switzer's book (if I remember right) yet I see no need for Hausdorffness in $Y$.
$\mathrm{QUESTION~1:~}$Is $Y$ Hausdorff really necessary?
Also, the reference I am currently using, Algebraic Topology from the Homotopical Viewpoint [Aguilar, Gitler, Prieto, Springer Universitext], exercice $1.3.4$ asks to show that for $X,Y,Z$ topological spaces with $X$ and $Y$ locally compact Hausdorff spaces, composition $\mathrm{Map}(X,Y)\times\mathrm{Map}(Y,Z)\rightarrow\mathrm{Map}(X,Z),(f,g)\mapsto g\circ f$ is continuous. Yet I'm pretty sure all you need is for $Y$ to be locally compact$\dots$
$\mathrm{QUESTION~2:~}$ Are all these extra conditions necessary?