Can a closed set in $\Bbb R$ be written in terms of open sets Is it possible to write any non-empty closed set in $\Bbb R$ as a combination of unions / intersection of open sets. Note that I don't demand just one union / intersection. I am happy with any combination (finite) of unions / intersection, but elements in each union/intersection must be open set. Any finite set, closed interval, cantor set can be written like this. 

For any set $A\subset \mathbb R$ define $B(A, r)=\left\{x\in\mathbb R:\text{dist(x,A) < r}\right\}$, where $\,\text{dist(x,A)} = \inf_{y\in A}\rho(x,y)$ with metric $\rho$. $B(A,r)$ is open for any $A$ and $r$. Now for every closed set $F\subset\mathbb R$
$$F = \bigcap_{n\in\mathbb N}B(F, \frac 1 n)$$
So every closed set in $\mathbb R$ is indeed an intersection of countably many open sets (so it is a $G_\delta$ set).
