What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)?



What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)?
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> Let $X$ be a space with $|X|= \aleph\_1$, let $\tau\_X= \lbrace U: X\setminus U \text{ is countable } \rbrace$. This space is CCC, but not separable.
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Proof: **$X$ is not separable:** for any countable set $A \subset X$, clearly, $U=X\setminus A$ is open and $U \cap A=\emptyset$.


**$X$ is CCC:** if $X$ has uncountable disjoint open sets $\lbrace \cal U\_\xi: \xi \in \aleph\_1\rbrace$. Pick one open set, for example, $U\_0$. Because $X \setminus U\_0$ is uncountable, it is a contradiction with $U\_0$ is open.


