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Noah Schweber
Noah Schweber
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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)$$\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 $c$.

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 $c$.

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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Noah Schweber
Noah Schweber
  • 20.7k
  • 10
  • 110
  • 332

Characterization of Stone-Cech compactifications

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 $c$.