Let $X$ be a compact Hausdorff space and $\mathcal{F}$ be a non-empty family of non-empty clopen subsets of $X$ with the finite intersection property. It's true that $\bigcap\mathcal{F}$ has non-empty interior?

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    $\begingroup$ It looks like the title and body ask different things. You don't mention the finite intersection property in the question body. $\endgroup$ – Ayman Hourieh Feb 7 '14 at 12:20
  • $\begingroup$ I'm sorry, I'll fix that. $\endgroup$ – user34870 Feb 7 '14 at 12:21

Take $X = \{ 0 \} \cup \{ \frac{1}{k} : k \geq 1 \}$ (as a subspace of $\mathbb{R}$), and for each $n$ define $F_n = \{ 0 \} \cup \{ \frac{1}{k} : k \geq n \}$.


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