Cartesian closed subcategories of compact Hausdorff topological spaces? The category of compact Hausdorff topological spaces is famously not cartesian closed. I was wondering how much more one has to assume to actually arrive to a cartesian closed category. 
For example, if we further assume total disconnectedness, i.e we end up with the category of Stone spaces, do we get cartesian closedness or still not? 
 A: Stone spaces are not cartesian closed, and I don't know of any interesting subcategory of compact Hausdorff spaces that is.  Typically, you get an interesting cartesian closed category related to compact Hausdorff spaces by allowing more spaces, not less: the problem is not that your spaces are too general, but simply that the natural topology on the set of all continuous maps between two spaces is almost never compact, so to have a mapping object you need to allow some non-compact spaces.
Here's a proof that Stone spaces aren't cartesian closed.  By Stone duality, if they were, then Boolean algebras would be cocartesian coclosed.  In particular, coproducts (aka tensor products) of Boolean algebras would distribute over arbitrary products.  This is false: if $B$ is an infinite Boolean algebra and $(A_i)$ is an infinite family of nontrivial Boolean algebras, the canonical map $B\otimes \prod A_i\to \prod B\otimes A_i$ is not surjective.  (You can prove this, for instance, by noting that any element of $B\otimes \prod A_i$ comes from $B_0\otimes \prod A_i$ for some finite subalgebra $B_0\subset B$.)
