# Are separability and ccc equivalent for closed subspaces of $\beta N$?

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 $$\beta \mathbb N$$?

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 $$\beta\mathbb{N}\setminus\mathbb{N}$$.