In general topology, an Fσ set (pronounced F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union). The complement of an Fσ set is a Gδ set. Fσ is the same as Σ 2 0 {\displaystyle \mathbf {\Sigma } _{2}^{0}} in the Borel hierarchy.
Examples Each closed set is an Fσ set. The set Q {\displaystyle \mathbb {Q} } of rationals is an Fσ set in R {\displaystyle \mathbb {R} } . More generally, any countable set in a T1 space is an Fσ set, because every singleton { x } {\displaystyle \{x\}} is closed. The set R ∖ Q {\displaystyle \mathbb {R} \setminus \mathbb {Q} } of irrationals is not an Fσ set. In metrizable spaces, every open set is an Fσ set. The intersection or union of finitely many Fσ sets is an Fσ set. Assuming the Axiom of countable choice, the union of countably many Fσ sets is an Fσ set. The set A {\displaystyle A} of all points ( x , y ) {\displaystyle (x,y)} in the Cartesian plane such that x / y {\displaystyle x/y} is rational is an Fσ set because it can be expressed as the union of all the lines passing through the origin with rational slope:
A = ⋃ r ∈ Q { ( r y , y ) ∣ y ∈ R } , {\displaystyle A=\bigcup _{r\in \mathbb {Q} }\{(ry,y)\mid y\in \mathbb {R} \},}
where Q {\displaystyle \mathbb {Q} } is the set of rational numbers, which is a countable set.
See also Gδ set — the dual notion. Borel hierarchy P-space, any space having the property that every Fσ set is closed
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