# Intersection of a closed and a dense subset of a Compact Hausdorff space be empty?

Suppose that $$X$$ is a compact Hausdorff space. Let $$\{X_i\}_{i \in \mathbb{N}}$$ be a sequence of dense open subsets of $$X$$. Let $$E$$ be a closed subset of $$X$$. Can it happen that $$\left(\cap_{i \in \mathbb{N}}X_i\right)$$ has empty intersection with $$E$$?

First by Baire Category theorem, $$\left(\cap_{i \in \mathbb{N}}X_i\right)$$ is dense in $$X$$. Moreover, since $$E$$ is closed, it is compact. If the intersection were empty, then $$E \subset \cup_{i \in \mathbb{N}}X\setminus X_i$$. But, this doesn't yield any contradiction.

A hint would be greatly appreciated.

Thanks for the help!!

Yes, the intersection can be empty. Suppose that $$X=[0,1]$$, with its usual topology. Take $$E=\{0\}$$ and $$X_n=(0,1]$$, for each natural $$n$$.
• You can also take a Cantor set $C$ and $X_n = [0,1] \setminus C$. Then $C$ is closed and uncountable! Since $C$ is nowhere dense, $X_n$ is dense. – p4sch Nov 21 '18 at 20:05