This question is asked at mathoverflow, but does not receive attention well.
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:
$$O \in \Sigma_1^0 \Longleftrightarrow \forall x \in O, \exists \varepsilon>0, V_{\varepsilon}(x) \subseteq O$$ where $V_{\varepsilon}(x)$ denotes the $\varepsilon$-neighbourhood of $x$. This definition is very useful as it allows us to visualize open set.
Question: Does there exist a definition of $F_{\sigma}$ set such that it is in terms of $\varepsilon?$