A question about Borel sets

Question

I have a question regarding the Borel sequence of real sets. Given the collection $\cal{F}_\sigma$ as sets that can be expressed as the countable union of closed sets and $\cal{G}_\delta$ as the collection of sets expressable as the countable intersection of open sets, I need to prove that all open sets are of type $\cal{F}_\sigma$ and all closed sets are of type $\cal{G}_\delta$.

I need a hint on a how to proceed. I know the Lindelof property of open sets on the real line, that is, every open set is a countable union of open interval and every open interval is a countable intersection of closed intervals. Is this the correct way to proceed?

Edit I have corrected the mistake in my question regarding open and closed sets and the classes $\cal{F}_\sigma$ and $\cal{G}_\delta$.

Answer 1

Let $A$ be a closed set. Since $\emptyset$ is a $\cal G_\delta,$ we may assume that $A\ne\emptyset.$ Then the function $$x\mapsto d(x,A)=\inf\{d(x,a):a\in A\}$$ is continuous, so for each $n\in\mathbb N$ the set $$U_n=\left\{x:d(x,A)\lt\frac1n\right\}$$ is open. I claim that $$A=\bigcap_{n\in\mathbb N}U_n.$$ The inclusion $A\subseteq\bigcap_{n\in\mathbb N}U_n$ is obvious. In the other direction, if $x\in\bigcap_{n\in\mathbb N}U_n,$ then $d(x,A)=0;$ since $A$ is closed, it follows that $x\in A.$

Answer 2

Any open interval is a countable union of closed intervals, i.e is an $F_{\sigma}$ set. Indeed: $$ (a,b)=\bigcup_{n=n_0}^{\infty}\left[a+\frac{1}{n},b-\frac{1}{n}\right] $$ for $n_0$ large enough to create such intervals. Similarly for rhe intervals $[a,\infty)$ and $(-\infty,b]$. Then by the Lindelof property you mentioned, any open set, as a countable union of open intervals, is $F_{\sigma}$. The 2nd part follows immediately by complements, but it could be showwn directly by a similar way.