Fσ set

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in France with F for fermé (French: closed) and σ for somme (French: sum, union). In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in France with F for fermé (French: closed) and σ for somme (French: sum, union). In metrizable spaces, every open set is an Fσ set. The complement of an Fσ set is a Gδ set. In a metrizable space, any closed set is a Gδ set. The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set. Fσ is the same as Σ 2 0 {displaystyle mathbf {Sigma } _{2}^{0}} in the Borel hierarchy. Each closed set is an Fσ set. The set Q {displaystyle mathbb {Q} } of rationals is an Fσ set. The set R ∖ Q {displaystyle mathbb {R} setminus mathbb {Q} } of irrationals is not a Fσ set. In a Tychonoff space, each countable set is an Fσ set, because a point x {displaystyle {x}} is closed. For example, 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: where Q {displaystyle mathbb {Q} } , is the set of rational numbers, which is a countable set.