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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 $\beta \mathbb N$?

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

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