Let $X$, $Y$, and $Z$ be locally-compact, complete, and separable metric spaces and suppose that $X$ is compact; all non-empty.


Consider the spaces $C(X,C(Y,Z))$ and $C(X\times Y,Z)$ both equipped with their uniform convergence on compact topologies. Does the map:
$$
C(X\times Y,Z)\ni f \to (x\mapsto [y\mapsto f(x,y)]) \in C(X,C(Y,Z)),
$$
define a homomorphism; and if so, is it uniformly continuous?


**References?:** *Where can I read more about this map...surely this has been studied somewhere in the general topology literature; probably in the mid 1950-1960s I expect?*



Relationship between $C(X\times Y,Z)$ and $C(X,C(Y,Z))$
In the context of functions from one space to another, a key-word here might by "currying", for the process of converting $x,y\to f(x,y)$ to $x\to (y\to f(x,y)$. In the case of spaces of holomorphic functions with the natural topology, verification is not hard.


In other contexts, isomorphisms $\operatorname{Hom}(X\otimes Y,Z)\approx \operatorname{Hom}(X,\operatorname{Hom}(Y,Z))$ are instances of the Eilenberg-Maclane *adjunction*. In simple algebraic situations (e.g., for abelian groups) the verification is easy.


Those instances suggest that we might want to characterize the various topologies in a fashion so as to make the assertion be correct... unless there is a "counter-example" that is important *not* to exclude for your purposes. That is, in effect, do you need/want the isomorphism to be correct, or not necessarily. :)


