Metric spaces have the following nice property:
Every closed set is a countable intersection of open sets. 

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> What other spaces have this property? Are there some nice known sufficient or necessary conditions on a topological space, for this property?
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(e.g. do locally compact Hausdorff spaces satisfy it? )

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The proof for metric spaces is very easy: 

Let $A\subseteq X$ be closed. For all $n\in \mathbb N$ define $$U\_n=\bigcup \_{a\in A} B(a,\frac{1}{n}).$$
Each $U\_n$ is open, and $A=\bigcap \_{n\in \mathbb N} U\_n$.