Is the following statement true, and if it is, does someone have a reference?

Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =Stone dual of the Boolean algebra of regular open sets) of $X$ is (naturally homeomorphic to) the projective limit of $\beta V$ for $V$ ranging over dense open subsets of $X$, where $\beta$ is the Stone-Čech compactification functor (and, of course, the maps $\beta V'\to \beta V$ are obtained from the inclusion $V'\to V$ by functoriality).

Since every $V$ and also every $\beta V$ has the same regular open algebra, hence the same Gleason space $E(X)$, there is at least a continuous map from $E(X)$ to the projective limit in question, and I am asking whether it is a homeomorphism commuting with the obvious maps from both of these spaces to $X$. I don't suppose it should be very difficult, but I'd prefer a reference.

I looked in Porter & Woods's book *Extensions and Absolutes of Hausdorff Spaces* with no success. There is, however, a related exercise 6Z (p.521) which describes $E(X)$ as as different projective limit (whose morphisms are finite-to-one; essentially by "splitting" finite systems of regular closed sets).

The projective limit of the $\beta V$ for $V$ open dense occurs in Fine, Gillman & Lambek's *Rings of Quotients of Rings of Functions*, but the words "absolute" or "Gleason" don't seem to appear anywhere.

**Edit:** It turns out that I didn't look carefully enough in Fine, Gillman & Lambek's book: the statement does appear there (even if the term "absolute" is not mentioned), by combining theorem 6.9 with theorem 11.15. I much prefer the proof that Adam Przeździecki gives below, however, because it is "direct" (describing the arrow on the spaces themselves) rather than "dual" (through maximal ideals and rings of functions), so it is more enlightening.

Commentationes Mathematicae Universitatis Carolinae24(1983), 233-236. It gives a description of the projective cover as the Stone-Čech compactification of the strongest topology subject to the condition that the identity map is irreducible. $\endgroup$ – Tomasz Kania Aug 25 '14 at 18:49