Well, this isn't a full answer, but I think it's worth posting.
We can identify a Hausdorff space with its poset of open sets because every ultrafilter converges uniquely to a point, so in fact, all of the "set data" of the space is contained in the poset of opens itself. We can define a lattice structure on this poset that takes the place of our algebra defined by intersections and unions. This is why the category of Hausdorff spaces is actually a category monadic over sets. Hausdorff spaces are actually totally described by their algebras, which seems pretty nifty to me.