# 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?

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

## 1 Answer

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