Does every open set contain a dense $F_{\sigma}$ subset?

Question

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set).

[ In my specific situation $U$ is of the form $\operatorname{int} f^{-1}(0)$, where $f$ is a continuous real-valued function on $X$, and $X$ is a Baire space (a sequence of dense open sets has a dense intersection), but I am not sure if it helps. ]

Is there a sequence $\{A_n\}_{n\in\mathbb{N}}$ of closed (in $X$) subsets of $U$ such that $\bigcup_{n\in\mathbb{N}} A_n$ is dense in $U$?

Of course, this is the case if $X$ is perfectly normal (which is equivalent to every open set being $F_{\sigma}$), or separable, but I hope a less restrictive assumption will suffice, e.g. normality.

Answer 1

Not in $\beta\mathbb{N}\setminus\mathbb{N}$: if $A$ is an $F_\sigma$-subset of $\operatorname{int}f^{-1}(0)$ then there is even a clopen set $C$ such that $A\subseteq C\subseteq \operatorname{int}f^{-1}(0)$. Of course this is only an example if the zero-set is not clopen but you can get an example by working on the countable set $N=\mathbb{N}^2$ the clopen sets determined by the vertical lines $V_n=\{n\}\times\mathbb{N}$ union up to a cozero set: let $f$ have value $2^{-n}$ on $V_n$. That cozero set is not clopen and for every closed-set $F$ contained in $\operatorname{int}f^{-1}(0)$ there is a function $h:\mathbb{N}\to\mathbb{N}$ such that $F$ is in the clopen set determined by $L_h=\{(m,n):n\le h(m)\}$. If $A$ is an $F_\sigma$ then we get a sequence $\langle h_n:n\in\omega\rangle$ of such functions. Define $h(m)=1+\max\{h_n(m):n\le m\}$. Then the clopen set determined by $L_h$ contains $A$ and is a subset of $\operatorname{int}f^{-1}(0)$.

We consider the behaviour of the extension of $f$ to $\beta N$ and its restriction to $N^*=\beta N\setminus N$ (all called $f$). The important thing to note is that if $X$ determines a clopen set, denoted $X^*$, in $N^*$ then $X^*\subseteq f^{-1}(0)$ iff $X\cap V_n$ is finite for all $n$. In general $X^*\cap Y^*=\emptyset$ iff $X\cap Y$ is finite, and in this case if $X\cap V_n$ is finite for all $n$ then $f[X^*]=\{0\}$. Furthermore: $X\cap V_n$ is finite for all $n$ iff $X\subseteq L_h$ for some $h$ as above. Finally: $N^*$ is compact and zerodimensional, so if $F$ is closed and contained in $\operatorname{int}f^{-1}(0)$ then there is a clopen set $C$ such that $F\subseteq C\subseteq \operatorname{int}f^{-1}(0)$.

See this note for a short introduction.