How about Parovicenko's Theorem ?

Assume CH. Let X be a compact zero-dimensional space without isolated points such that:

1. X in an $F$-space (that is, disjoint open $F_\sigma$ sets have disjoint closures).

2. $X$ is an almost $P$-space (that is, every non-empty $G_\delta$ set has non-empty interior).

3. $X$ has a base of cardinality continuum.

Then $X$ is homeomorphic to $\beta \mathbb{N} \setminus \mathbb{N}$

Parovicenko's characterization of $\beta \mathbb{N} \setminus \mathbb{N}$ is actually equivalent to CH by a result of van Douwen and van Mill.

But now that I think about it, Parovicenko's theorem can be stated using essentially two hypotheses if we translate everything in the language of Boolean Algebras using Stone duality...

How about Parovicenko's Theorem ?

Assume CH. Let X be a compact zero-dimensional space without isolated points such that:

1. X is an $F$-space (that is, disjoint open $F_\sigma$ sets have disjoint closures).

2. $X$ is an almost $P$-space (that is, every non-empty $G_\delta$ set has non-empty interior).

3. $X$ has a base of cardinality continuum.

Then $X$ is homeomorphic to $\beta \mathbb{N} \setminus \mathbb{N}$

Parovicenko's characterization of $\beta \mathbb{N} \setminus \mathbb{N}$ is actually equivalent to CH by a result of van Douwen and van Mill.

It's interesting that Parovicenko's theorem is equivalent to a boolean-algebraic statement which uses only two essential hypotheses...

How about Parovicenko's Theorem ?

Assume CH. Let X be a compact zero-dimensional space without isolated points such that:

1. X in an $F$-space (that is, disjoint open $F_\sigma$ sets have disjoint closures).

2. $X$ is an almost $P$-space (that is, every non-empty $G_\delta$ set has non-empty interior).

3. $X$ has a base of cardinality continuum.

Then $X$ is homeomorphic to $\beta \mathbb{N} \setminus \mathbb{N}$

Parovicenko's characterization of $\beta \mathbb{N} \setminus \mathbb{N}$ is actually equivalent to CH by a result of van Douwen and van Mill.

But now that I think about it, Provicenko's theorem can be stated using essentially two hypotheses if we translate everything in the language of Boolean Algebras using Stone duality...

How about Parovicenko's Theorem ?

Assume CH. Let X be a compact zero-dimensional space without isolated points such that:

1. X in an $F$-space (that is, disjoint open $F_\sigma$ sets have disjoint closures).

2. $X$ is an almost $P$-space (that is, every non-empty $G_\delta$ set has non-empty interior).

3. $X$ has a base of cardinality continuum.

Then $X$ is homeomorphic to $\beta \mathbb{N} \setminus \mathbb{N}$

Parovicenko's characterization of $\beta \mathbb{N} \setminus \mathbb{N}$ is actually equivalent to CH by a result of van Douwen and van Mill.

But now that I think about it, Parovicenko's theorem can be stated using essentially two hypotheses if we translate everything in the language of Boolean Algebras using Stone duality...