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The compact Hausdorff space $X = \beta\mathbb{N}$ is another example. Every regular open subset is the closure of a subset of $\mathbb{N}$ and there are only $\frak{c}$ such subsets but $X$ has $2^{\frak{c}}$ points, and for each such point $p$, the set $X \setminus \{p\}$ is an open set.