# Q: Does every open subset of a second-countable topological space is $F_\sigma$?

I'm trying to solve the following problem:

Let $(X,T)$ be a second-countable, normal topological space, and let $U \subset X$ be an open set. Show that there exists a continuous function $f:X\rightarrow[0,1]$ such that $f(x) \neq 0$ iff $x \in U$.

Now, I know that in a metric space, every open set is $F_\sigma$. Does this holds also for a general second-countable space? I've got a hunch that the answer is positive, but I couldn't prove it.

If so, solving the problem would be easy, since $U=\bigcup_{i=1}^\infty F_i$, where each $F_i$ is closed, and so there exists a closed set $F$ such that $F \subset U \subset X$, and by applying Urysohn's lemma to $U^c$ and $F$ we are done.

Yes, this is true. Suppose $B_n, n \in \mathbb{N}$ is a countable base for $X$, and $U$ is open. Then for each $x \in U$ we pick by regularity some open $V_x$ such that $x \in V_x \subseteq \overline{V_x} \subseteq U$. Then pick some $n(x)$ with $x \in B_{n(x)} \subseteq V_x$. Then all $\cup \{\overline{B_{n(x)}}: x \in U \}$ is equal to $U$ (all $x$ in $U$ are in their own $B_{n(x)}$; all of these have closures inside $U$ by construction, so their union is exaclty $U$) and shows $U$ is an $F_\sigma$ (as we have countably many $n(x)$ (many $x$ will give the same $n(x)$, but that does not matter, we have countably many (at most) different closed sets).