Fσ set - Alchetron, The Free Social Encyclopedia

In mathematics, an F_σ 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 F_σ set. The complement of an F_σ set is a G_δ set. In a metrizable space, any closed set is a G_δ set.

The union of countably many F_σ sets is an F_σ set, and the intersection of finitely many F_σ sets is an F_σ set. F_σ is the same as Σ 2 0 in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

The set Q of rationals is an F_σ set. The set R ∖ Q of irrationals is not a F_σ set.

In a Tychonoff space, each countable set is an F_σ set, because a point x is closed.

For example, the set A of all points ( x , y ) in the Cartesian plane such that x / y is rational is an F_σ 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 } ,

where Q , is the set of rational numbers, which is a countable set.

References

Fσ set Wikipedia

(Text) CC BY-SA