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. 



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


