The Stone space functor we’ve been working with sends Boolean algebras to topological spaces. Specifically, it sends them to compact Hausdorff spaces. There’s another functor floating around, of course, though it might not be the one you expect.
The clue is in our extended result. Given a topological space we define to be the Boolean algebra of all clopen subsets. This functor is contravariant — given a continuous map , we get a homomorphism of Boolean algebras sending the clopen set to its preimage . It’s straightforward to see that this preimage is clopen. Another surprise is that this is known as the “Stone functor”, not to be confused with the Stone space functor .
So what happens when we put these two functors together? If we start with a Boolean algebra and build its Stone space , then the Stone functor applied to this space gives us a Boolean algebra . This is, by construction, isomorphic to itself. Thus the category is contravariantly equivalent to some subcategory of . But which compact Hausdorff spaces arise as the Stone spaces of Boolean algebras?
Look at the other composite; starting with a topological space , we find the Boolean algebra of its clopen subsets, and then the Stone space of this Boolean algebra. We also get a function . For each point we define the Boolean algebra homomorphism that sends a clopen set to if and only if . We can see that this is a continuous map by checking that the preimage of any basic set is open. Indeed, a basic set of is for some clopen set . That is, . Which functions of the form are in ? Exactly those for which . Since is clopen, this preimage is open.
Two points and are sent to the same function if and only if every clopen set containing also contains , and vice versa. That is, and must be in the same connected component. Indeed, if they were in different connected components, then there would be some clopen containing one but not the other. Conversely, if there is a clopen that contains one but not the other they can’t be in the same connected component. Thus this map collapses all the connected components of into points of .
If this map is a homeomorphism, then no two points of are in the same connected component. Thus each singleton is a connected component, and we call the space “totally disconnected”. Clearly, such a space is in the image of the Stone space functor. On the other hand, if , then , and so this is both a necessary and a sufficient condition. Thus the “Stone spaces” form the full subcategory of , consisting of the totally disconnected compact Hausdorff spaces. Stone’s representation theorem shows us that this category is equivalent to the dual of the category of Boolean algebras.
As a side note: I’d intended to cover the Stone-Čech compactification, but none of the references I have at hand actually cover the details. There’s a certain level below which everyone seems to simply assert certain facts and take them as given, and I can’t seem to reconstruct them myself.