We call a topological space $X$ extremely disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremely disconnected spaces are totally disconnected in the sense that their connected components are singletons.

Suppose $\mathbb{Z}_p = \varprojlim_{n \ge 1} \mathbb{Z}/p^n\mathbb{Z}$ is the ring of $p$-adic integers, viewed as a closed subspace of the compact Hausdorff space $\prod_{n \ge 1} \mathbb{Z}/p^n\mathbb{Z}$, where the latter is equipped with the product topology and each factor $\mathbb{Z}/p^n\mathbb{Z}$ is discret. Then it is well-known that $\mathbb{Z}_p$ is totally disconnected. But I can't answer the seemingly naive question: is $\mathbb{Z}_p$ extremely disconnected?

Following the traditions of operator algebraists, we call a compact Hausdorff extremely disconnected space a stonean space. And we call a stonean space hyperstonean if it admits enough normal measures---a technical condition which characterises hyperstonean spaces exactly as the spectrum of abelian von Neumann algebras. A series of possibly less naive questions are: if $\mathbb{Z}_p$ is stonean (which is the same as its extreme disconnectedness), is it hyperstonean? And if it is hyperstonean, can we associate some natural Hilbert space (like $L^2(\mathbb{Z}_p, \mu)$, where $\mu$ is the normalized Haar measure on $\mathbb{Z}_p$ viewed as a compact abelian topological group), and an abelian von Neumann algebra (hopefully the algebra $C(\mathbb{Z}_p)$ of complex-valued continuous functions on $\mathbb{Z}_p$) acting on this Hilbert space, such that $\mathbb{Z}_p$ is exactly the spectrum of this latter von Neumann algebra?

Finally, a perhaps stupid and somewhat less well-formulated question: do we have some "nice" examples of stonean, even hyperstonean spaces, that are infinite, and are not constructed as the spectrums of their associated operator algebras (of course they do appear as such spectrums, I mean one describe the relevant topologies without referring first to the algebra of functions on it)?

I ask these questions because as far as I know, unlike the widely spread notion of totally disconnected spaces (or $0$-dimensional spaces), (hyper)stonean spaces seems to be only of a minor interest to operator algebraists as spectrums of abelian von Neumann algebras (except Gleason's theorem characterising them as the projective object in the category of compact Hausdorff spaces). And the only way of producing nontrivial stonean spaces that I am aware of is taking the spectrum of some algebras. If $\mathbb{Z}_p$ does turn out to be stonean, it would be a cute concrete example in my humble opinion.