Closed set as a countable intersection of open sets Let's take a metric space. Then any closed set can be written as a countable intersection of open sets.
How can I prove that?

Hint: Let $A\subseteq X$ be closed. For $\epsilon > 0$ let 
$$ U_\epsilon(A) = \{x \in X \mid \def\dist{\mathop{\rm dist}}\dist(x,A) < \epsilon \}$$
where $\dist(x, A) := \inf_{y \in A} d(x,y)$. What can you say about the sets $U_\epsilon(A)$, what is $\bigcap_{\epsilon > 0} U_\epsilon(A)$?
