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Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not sequentially compact, has a basis of clopen sets, etc. My question is the following: is there a "nice" characterization of the spaces $Y$ which are homeomorphic to the Stone-Cech compactification of a discrete space? Certainly, the term "nice" is vague; I have in mind characterizations only using terms from a standard text on point-set topology, but I would consider as an answer to this question really any nontrivial characterization of Stone-Cech compactifications.

I am particularly interested in nice characterizations that require some set theory, such as "assuming $V=L$, $Y$ is homeomorphic to $\beta X$ for some discrete $X$ iff $Y$ is compact, not sequentially compact, and has a basis of clopen sets" (although I'm certain that statement is extremely false), and I would especially like to know whether there are two incompatible strong set-theoretic assumptions which yield distinct nice characterizations. The only relevant result I know is along these lines: in 1963, Parovicenko showed that assuming CH, the only Parovicenko space (which has a long but elementary definition*) is $\beta\mathbb{N}-\mathbb{N}$; this can be molded into a characterization of $\beta\mathbb{N}$, assuming CH, but says nothing about whether a space is the Stone-Cech compactification of a discrete space of uncountable cardinality. In 1978, van Douwen and van Mill showed that CH was necessary. One more concrete sub-question I have, then, is:

Does Parovicenko's result generalize in some way to characterize Stone-Cech compactifications of larger discrete spaces? If so, how much set theory is needed - is GCH enough?

(One very tempting way to try to rephrase Parovicenko's result is to define "$\kappa$-Parovicenko space" by taking the definition of Parovicenko space and replacing the "weight $c$" condition with "weight $2^\kappa$," and then claiming that - assuming GCH - every $\kappa$-Parovicenko space is homeomorphic to $\beta X-X$ for a discrete space $X$ of cardinality $\kappa$. However, I see absolutely no reason to believe this. A sub-subquestion: is this statement obviously false?)


*For completeness, a Parovicenko space is a topological space which is compact and Hausdorff, has no isolated points, has no nonempty $G_\delta$ set with empty interior, has no two disjoint $F_\sigma$ sets with non-disjoint closures, and has weight $c=2^{\aleph_0}$ - that is, every basis has cardinality $\ge c$, and there is some basis with cardinality $c$.

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I think this might be correct (i would have to check it...) : X is the stone-Cech compactification of a discret space if and only if : X is compact, Hausdorf, extremally disconected (en.wikipedia.org/wiki/Extremally_disconnected_space) and has a dense subset open "open point" (point x such that {x} is open ). is this interesting for you ? –  Simon Henry Sep 23 '12 at 7:21
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up vote 9 down vote accepted

I confirme my comment :

$X$ is the stone-cech compactification of a discrete space if and only if $X$ is compact, haussdorf, extremally disconected, and has a dense set of open points.

here is a sketches of the proof :

If X is a stone-chech compactification of a discret set Y, then it is clear that X is compact, hausdorf, the point of Y form a dense set of open points, and it is well know that X is extremally disconected.

Asume now that $X$ is a topological space satisfying all those hypothesis.

Let $Y$ be the set of open point of $X$.

It's a routine to check to see that the map which to a subset $P$ of $Y$ associate it's closure in $X$, and the map which to a clopen of $X$ associate it's intersection with $Y$, are reciprocal bijection between the parts of $Y$ and the clopen set of $X$.

considere now a point $x \in X$, then {x} is the intersection of clopen set containing $X$, and the set of clopen of $X$ containing $x$ correspond through the previous bijection to an ultrafilter on $Y$.

After that, consider an ultrafilter $\mathcal{F}$ on $Y$, you can see that $\displaystyle \bigcap_{P \in \mathcal{F}} \overline{P} $ is a singleton (it contains a point because it is an intersection of non-empty compact, and it can't contain two point because of the properties ultrafilter).

those two application will induce an homeomorphism between $X$ and the space of ultrafilter of $Y$.

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+1, this is interesting. Andrew Gleason proved that the projective objects in compact Hausdorff spaces (i.e., retracts of free objects, i.e., retracts of Stone-Cech compactifications of discrete spaces) are exactly the compact, Hausdorff, extremally disconnected spaces (this result can be found in Johnstone's Stone Spaces). –  Todd Trimble Sep 23 '12 at 12:10
    
actualy my idea come more from the fact that compact, hausdorf, extremally disconected space (stonean space) are exactly the stone-Cech compactification of boolean local (I think it's in Johnstone's Stone Space too, isn't it ? ). So i just had to add an hypothesis of existence of "points" for the local of clopen to be able to conclude that the starting boolean local was a discret space. But i guess those two result are closely related. –  Simon Henry Sep 23 '12 at 13:02
    
Ah, I think I see. So the compact Hausdorff extremal disconnectedness of $X$ means the clopens of $X$ form a complete Boolean algebra $B$. The open points are atomic elements in $B$, and the density should imply that every clopen is a join in $B$ of such atomic elements. $B$ is therefore a complete atomic Boolean algebra, and thus of the form $PS$ for some set $S$. The Stone space of $B$ (which is $X$) is thus the Stone space of $PS$, which is $\beta(S)$ essentially by definition. Thanks! –  Todd Trimble Sep 23 '12 at 14:58
    
This is very nice! Thanks a lot. –  Noah S Sep 23 '12 at 19:31
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