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.