Corollary 2

Let $X, Y, Z$ be three topological spaces such that $X$ is Hausdorff and $Y$ is locally compact. Then the restriction to $C(X \times Y; Z)$ of the canonical bijection $\mathscr{F}(X \times Y; Z) \to \mathscr{F}(X; \mathscr{F}(Y; Z))$ (Set Theory, R, § 4, no. 14) is a homeomorphism of $C_c(X \times Y; Z)$ onto $C_c(X; C_c(Y; Z))$.

This restriction is certainly a bijection

by Theorem 3; it remains therefore to be shown that the compact-open topology on $\mathscr{C}(X \times Y; Z)$ is the inverse image under $\rho$ of the compact-open topology on $\mathscr{C}(X; \mathscr{C}_c(Y; Z))$. Since the sets $T(K, U)$, where $K$ is a compact subset of $Y$ and $U$ is an open subset of $Z$, form a subbase of the topology of $\mathscr{C}_c(Y; Z)$, it follows from Remark 2 that the topology of $\mathscr{C}_c(X; \mathscr{C}_c(Y; Z))$ is generated by the sets of the form $T(J, T(K, U))$, where $K$ and $U$ are as above and $J$ is a compact subset of $X$. Now the image of $T(J, T(K, U))$ under $\rho$ is precisely $T(J \times K, U)$, and is therefore an open set; so we have shown that $\rho$ is continuous. To prove that $\rho$ is a homeomorphism, we note first that the sets of the form $J \times K$ in $X \times Y$ (where $J$ is a compact subset of $X$, and $K$ is a compact subset of $Y$) satisfy the condition (R) of Remark 2: for if $L$ is a compact subset of $X \times Y$ and $V$ is a neighbourhood of $L$ in $X \times Y$, the projections $M = \operatorname{pr}_1(L)$, $N = \operatorname{pr}_2(L)$ are compact, since $X$ and $Y$ are Hausdorff and $V \cap (M \times N)$ is a neighbourhood of $L$ in the compact space $M \times N$, so that every point of $L$ has a neighbourhood in $M \times N$ of the form $J \times K \subset V$, where $J \subset M$ and $K \subset N$ are compact; since $L$ can be covered by a finite number of these neighbourhoods, the assertion is proved. Therefore sets of the form $T(J \times K; U)$, where $J$ is a compact subset of $X$, $K$ a compact subset of $Y$ and $U$ an open subset of $Z$, generate the topology of $\mathscr{C}_c(X \times Y; Z)$. But we have already seen that the image of $T(J \times K, U)$ under $\rho$ is the open set $T(J, T(K, U))$ in $\mathscr{C}_c(X; \mathscr{C}_c(Y; Z))$; hence $\rho$ is a homeomorphism.

Note that if in addition $Z$ is assumed to be uniformizable, Corollary 2 is a trivial consequence of § 1, no. 4, Proposition 2.