$F_\sigma$ and $G_\delta$ subsets of $\mathbb{R}$

Let $A \subseteq \mathbb{R}$, with open and closed sets as in Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen.

• $A$ is an $F_\sigma$ set when there is a sequence $(F_n)_{n \in \mathbb{N}}$ of closed subsets of $\mathbb{R}$ with $A \;=\; \bigcup_{n \in \mathbb{N}} F_n .$
• $A$ is a $G_\delta$ set when there is a sequence $(V_n)_{n \in \mathbb{N}}$ of open subsets of $\mathbb{R}$ with $A \;=\; \bigcap_{n \in \mathbb{N}} V_n .$

The letters are the traditional ones: $F$ for fermé with $\sigma$ for somme, $G$ for Gebiet with $\delta$ for Durchschnitt.

The two classes are exchanged by complementation. $A$ is $F_\sigma$ if and only if $\mathbb{R} \setminus A$ is $G_\delta$. If $A = \bigcup_n F_n$ with each $F_n$ closed, then $\mathbb{R} \setminus A = \bigcap_n (\mathbb{R} \setminus F_n)$ by De Morgan, and each $\mathbb{R} \setminus F_n$ is open by the definition of closedness (Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen); the converse is the same computation read backwards, using that the complement of an open set is closed, which is again Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen.

Every closed set is $F_\sigma$ and every open set is $G_\delta$, by the constant sequence $F_n := A$, respectively $V_n := A$. As with Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$, an at most countable family (Finite, countably infinite, countable, uncountable) may always be presented as a sequence: a finite list $F_0, \dots, F_m$ of closed sets is extended by $F_n := F_m$ for $n > m$, and a finite list of open sets likewise, so nothing is lost by indexing over $\mathbb{N}$.

• The classes are genuinely larger than the closed and the open sets. $\mathbb{Q}$ is $F_\sigma$ and is neither open nor closed, and the irrationals are $G_\delta$ and neither open nor closed; both computations are in $\mathbb{Q}$ is $F_\sigma$, meager and not $G_\delta$, while the irrationals are $G_\delta$, residual and not $F_\sigma$. That $\mathbb{Q}$ is not also $G_\delta$ is the first genuinely hard fact about these classes and needs the Baire category theorem (Baire category in $\mathbb{R}$, by nested intervals with canonically chosen rational endpoints: a countable intersection of dense open sets is dense, so $\mathbb{R}$ is not a countable union of nowhere dense sets).

• Why the algebra of open sets is not enough. Arbitrary unions and finite intersections of open subsets of $\mathbb{R}$ are open, and dually for closed sets gives that a finite intersection of open sets is open and a finite union of closed sets is closed. The definitions above are exactly what one gets by relaxing "finite" to "countable" once, and the point of the whole notion is that the relaxation is proper: a countable intersection of open sets need not be open, which is $\bigcap_k (-1/k, 1/k) = \{0\}$ is not open.

• Nothing here is a measure-theoretic notion. $F_\sigma$ and $G_\delta$ are defined from the topology of $\mathbb{R}$ alone and are used on this page to say precisely how far $\mathbb{Q}$ and its complement sit from being closed or open. They cut across Measure zero (a countable cover by intervals of total length below every $\varepsilon$) and content zero (a finite such cover) completely: $\mathbb{Q}$ is $F_\sigma$ and has measure zero (Every at most countable subset of $\mathbb{R}$ has measure zero), while the Smith-Volterra-Cantor set is closed, hence $F_\sigma$, and does not (The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero).