My question arose after I read Topologies on spaces of continuous functions of Martin Escardo and Reinhold Heckmann. Terminology of the question is similar to note. I avoid here of definitions of all terms, they can be wived in a note I mentioned.
Suppose $X$ is a general topological space. Let $S$ denotes a Sierpiński space. Note that we can identify $C(X,S)$ with $\mathcal{O}X$, a set of opens of $X$. Suppose $T$ is a exponential topology on $C(X,S)$ we can identify it with a topology on $\mathcal{O}X$. Givet topological space $Y$ we can induce a topology on $C(X,Y)$ as a topology generated by subbasic open sets $$T(O,V)=\{f\in C(X,Y)\mid f^{-1}(V)\in O\}, $$ $O$ ranges over $T$, $V$ ranges over $\mathcal{O}Y$.
Now let us change a category, that is restrict it. Consider only compact Hausdorff spaces. It is known that exponentiable compact Hausdorff spaces are exactly finite Hausdorff spaces see:Cagliari, Mantovani Theorem 2.6
My question is does compact Hausdorff exponentiable topology on $C(X,S)$ induce a compact hausdorf topology on $C(X,Y)$ as above? And in case that it does, that is likely the case, how it can be seen?
In particular does compactness and Hausdorffness of $T$ on $C(X,S)$ imply compactness of induced topology on $C(X,Y)?$
