Definition of $F_{\sigma}$ sets in terms of $\varepsilon$?

Question

Let $X$ be a metric space.

In Borel hierarchy, $\Sigma_{1}^0$ is the set of all open sets in $X$ while $\Pi_{1}^0$ is the set of all closed sets in $X.$ Then at next level, one has $\Sigma_{2}^0 = \{ \cup_{n \in \mathbb{N}} A_n : A_n \in \Pi_1^0 \}$, that is, elements of $\Sigma_2^0$ are $F_{\sigma}$ sets.

In $\Sigma_1^0$, its elements can be described in the following manner: $O \in \Sigma_1^0 \Longleftrightarrow$ $\forall x \in O, \exists \varepsilon>0$ such that $V_{\varepsilon}(x) \subseteq O,$ where $V_{\varepsilon}(x)$ denotes the $\varepsilon$-neighbourhood of $x$. This definition is very useful as it allows us to visualize open set.

Question: How to visualize a $F_{\sigma}$ set? In other words, what is the definition of $F_{\sigma}$ set in terms of $\varepsilon?$

Answer 1

Suppose that $(X,d)$ is a complete metric space. Then a subset $G\subseteq X$ is a $G_{\delta}$-set precisely when $G$ can be given a complete metric which induces the subspace topology on $G$. The notions of completeness and compatibility can easily be written in terms of $\epsilon,\delta$ and a new metric. I therefore just visualize $F_{\sigma}$ sets as the sets whose complements can be given compatible complete metrics.