2
$\begingroup$

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable family of pairwise disjoint open sets.

Are separability and ccc equivalent for a closed subset of $\beta \mathbb N$?

$\endgroup$
5
$\begingroup$

No. There is a compactification of $\mathbb{N}$ whose remainder, $K$, is ccc non-separable. So there is a continuous surjection $f$ from $\beta\mathbb{N}\setminus\mathbb{N}$ onto $K$; take a closed subset $F$ such that $f$ is irreducible on $F$. Then $F$ is a ccc non-separable subspace of $\beta\mathbb{N}\setminus\mathbb{N}$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.