Here's one I learned from Todd Trimble. Giving a set $X$ the structure of a compact Hausdorff space is the same as equipping $X$ with $J$-ary operations $X^J \to X$ for every set $J$, one for each ultrafilter $P$ on $J$, corresponding to the $P$-limit of a $J$-tuple of elements of $X$, with the appropriate compatibility relations.