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?
[Math] Closed set as a countable intersection of open sets
metric-spaces
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Best Answer
Let $A\subseteq X$ be closed. For all $n\in \mathbb N$ define $$U_n=\bigcup _{a\in A} B(a,\frac{1}{n}).$$ $U_n$ is open as a union of open balls. We prove that $A=\bigcap _{n\in \mathbb N} U_n$.
Clearly $A \subseteq \bigcap _{n\in \mathbb N} U_n$.
To prove $A \supseteq \bigcap _{n\in \mathbb N} U_n$ we take $x\notin A$ and show that $x\notin \bigcap U_n$.
Since $A$ is closed, $A^C$ is open, therefore $\exists n \in \mathbb N$ such that $B(x,\frac{1}{n}) \cap A=\emptyset$. That is, for all $a\in A$: $a\notin B(x,\frac{1}{n})$; and thus for all $a\in A$: $x\notin B(a,\frac{1}{n}) \Longrightarrow x\notin \bigcup_{a\in A} B(a,\frac{1}{n}) \Longrightarrow x\notin U_n \Longrightarrow x\notin \bigcap U_n$.