What are the open sets in $\mathbb{Q}$ wrt euclidean topology?
Basically I want to know whether an arbitrary open set in a metric space is countable union of closed sets or not, with or without depending on the locally compactness of the metric space.
A space in which every open set is an $F_{\sigma}$ set ( a countable union of closed sets) is called a perfectly normal space. Metric spaces are perfectly normal.
Let $(X,d)$ be a metric space and let $U\subset X$ be open. For $n\in \mathbb N$ let $$V_n=\{p\in X: \forall q\in X \backslash U\;(\;d(p,q)\geq 1/n\;)\}.$$ (i). For each $p\in U$ there exists $n\in \mathbb N$ with $B_d(p,1/n)\subset U,$ so $p\in V_n$ for some $n\in \mathbb N.$ $$\text {So }\quad U\subset\cup_{n\in \mathbb N}V_n.$$ (ii). If $p\in X$ \ $U$ then $p\not \in V_n $ for any $n$, because there does exist $q\in X$ \ $U$ with $d(p,q)<1/n,$ namely $q=p.$ $$ \text {So } \quad U\supset \cup_{n\in \mathbb N}V_n.$$ $$\text {(iii). Therefore }\quad U=\cup_{n\in \mathbb N}V_n.$$ Each $V_n$ is closed. By contradiction, suppose $(p_j)_j$ is a sequence in $V_n$ converging to $x\not \in V_n.$ Then there exists $q\in X$ \ $U$ with $r=d(p,q)<1/n.$ But there exists $j$ with $d(p_j,x)<((1/n)-r)/2,$ implying $$d(p_j,q)\leq d(p_j,x)+d(x,q)<((1/n)-r)/2+r=((1/n)+r)/2<1/n,$$ contradicting $p_j\in V_n.$
