TITLE: Sufficient conditions for $M(X \times Y, Z)$ to be homeomorphic to $M(X, M(Y, Z))$
QUESTION [8 upvotes]: Let $Y^X$ denote the set of all functions $f: X \to Y$. If $X$ and $Y$ are topological spaces, let $M(X,Y)$ denote the set of all continuous maps $f: X \to Y$, endowed with the compact-open topology. Consider the functions $\varphi: Z^{X \times Y} \to (Z^Y)^X$ and $\psi: (Z^Y)^X \to Z^{X \times Y}$ defined by $\varphi(f)(x)(y) = f(x,y)$ and $\psi(g)(x,y) = g(x)(y)$, for all $f \in Z^{X \times Y}$, $g \in (Z^Y)^X$, $x \in X$ and $y \in Y$. Is easy to verify that $\varphi$ and $\psi$ are mutual inverse functions.
In Aguilar's Algebraic Topology from a Homotopical Viewpoint, we find the following results:
1.3.1. Theorem. If $X$, $Y$ and $Z$ are topological spaces with $Y$ Hausdorff and locally compact, then the image of $M(X \times Y, Z)$ under $\varphi$ is $M(X, M(Y,Z))$, whereas the image of $M(X, M(Y,Z))$ under $\psi$ is $M(X \times Y, Z)$. Therefore, the restrictions $\varphi \restriction M(X \times Y, Z)$ and $\psi \restriction M(X, M(Y,Z))$ are inverse bijections between $M(X \times Y, Z)$ and $M(X, M(Y,Z))$.
1.3.2. Theorem. If $X$, $Y$ and $Z$ are topological spaces such that $X$ and $Y$ are Hausdorff and $Y$ is locally compact, then $\varphi \restriction M(X \times Y, Z)$ is a homeomorphism onto $M(X, M(Y,Z))$.
My doubt is about the last theorem above. I cannot see in its proof where the condition of $X$ being Hausdorff come into play. Is this condition really important? Here is a picture showing this proof and its context in the book:

To confuse me further, Dugundji's Topology presents the following:
5.3 Theorem (chapter XII). If $X$, $Y$ and $Z$ are topological spaces with $Y$ Hausdorff locally compact and $X$, $Z$ arbitrary, then $\varphi \restriction M(X \times Y, Z)$ is a homeomorphism onto $M(X, M(Y,Z))$.
I have adapted the terminology and notation in this statement, but the word "arbitrary" really appears there in the book. So, is the condition of $X$ being Hausdorff dispensable? Why did Aguilar et al. include this condition in the statement of Theorem 1.3.2?

REPLY [2 votes]: The assumption that $X$ is Hausdorff is already used in the first step of the displayed proof—in the claim that the sets of the form “$(U^L)^K$” form a subbasis.  Said claim follows from Lemma~XII.5.1 (a) of Dugundji’s Topology. Alternatively, see Tammo tom Dieck's Topologie, Satz 11.1.
Note that, in Topology, Dugundji implicitly assumes all spaces to be Hausdorff! This is mentioned in the introduction. There is also a reminder at the very beginning of section XII.6.
That said, I do not know whether the assumption on $X$ is essential or whether it just makes for a more convenient proof.