While studying for an exam i came across the problem:
Show that Map$(X,Y×Z)$ is homeomorphic to Map$(X,Y) ×$ Map$(X,Z)$. Where each is the space of continuous functions given the Compact Open Topology.
I have found there is a map $M(X,Y×Z) \to M(X,Y) × M(X,Z)$ which is bijective and continuous.
I can prove that the inverse function is continuous when $X$ is Hausdorff, But not in general. Can anyone provide me with a proof or reference that works in the general case?
For reference here is my proof when $X$ is Hausdorff.
Let $F:M(X,Y) ×M(X,Z) \to M(X,Y×Z)$. Let $A= S(K,U)$ be a pre-basis element. And let $(f,g)$ be an element of $F^{-1}(A)$. For each $x$ in $K$, find an open set $V$, so that $(f × g)(\bar{V} \cap K)$ is contained in a neighborhood $N × W \subseteq U$. Cover $K$ by finitely many such sets $V_1,...V_n$, and let $N_i × W_i$, be there corresponding sets. Then let $O$ be the intersection of all the $S(\bar{V_i},N_i) × S(\bar{V_i},W_i)$. Then $O$ is open and $(f,g )\in O \subseteq F^{-1}(A)$. So $F$ is continuous.
