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.

  • 2
    $\begingroup$ Your proof basically relies on the local compactness of $\bar K$, so it also holds when $X$ is locally compact. More generally, it works when $X$ is a space such that we have a cont. map $ε_Y:M(X,Y)×X→Y,(g,x)↦g(x)$ and a bijection $$\mathbf{Top}(W,M(X,Y))≅\mathbf{Top}(W×X,Y)$$ sending $f:W→M(X,Y)$ to $ε_Y(f×1_X)$. Then we have a sequence of isomorphisms$$\begin{align} \mathbf{Top}(W,M(X,Y×Z))&≅\mathbf{Top}(W×X,Y×Z)\\ &≅\mathbf{Top}(W×X,Y)×\mathbf{Top}(W×X,Z)\\ &≅\mathbf{Top}(W,M(X,Y))×\mathbf{Top}(W,M(X,Z))\\ &≅\mathbf{Top}(W,M(X,Y)×M(X,Z)) \end{align}$$This would imply that your map is an iso $\endgroup$ Jun 1, 2015 at 15:35


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