I would like to show precisely what I have stated in the title (assuming that it is correct; I have reason to suspect it is, thanks to a tricky past exam paper I'm trying to surmount); namely, that given any 2 compact Hdf topological spaces $T_1$ and $T_2$, the class of continuous functions between them forms a set (rather than, say, a class which is too large to be a set).
I suspect this result might be obvious in a much more general sense, and just wanted to check my thinking: is it actually valid to say that for any function between 2 fixed sets $S_1$ and $S_2$, the function is simply a set of ordered pairs $(a,f(a))$ and therefore contained in the power set of the union $S_1 \cup S_2$, so is again a set (by standard set-theoretic axioms for power sets and unions)?