Family of clopen with FIP in a compact space 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? 

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