My question arose after I read [Topologies on spaces of continuous functions][1] of Martin Escardo and Reinhold Heckmann. Terminology of the question is similar to note.

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  $$T(O,V)=\{f\in C(X,Y)|f^{-1}(V)\in O\}, $$ $O$ 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][2] 


  [1]: http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf
  [2]: http://ac.els-cdn.com/016686419190009B/1-s2.0-016686419190009B-main.pdf?_tid=3349c714-032a-11e7-a61a-00000aab0f6b&acdnat=1488886724_8239249cb9964d6052dd6303bc358fc2

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 $C(X,Y)?$