# Fσ set

(Redirected from F sigma set)

In mathematics, an Fσ set (said 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 $\mathbf {\Sigma } _{2}^{0}$ in the Borel hierarchy.

## Examples

Each closed set is an Fσ set.

The set $\mathbb {Q}$  of rationals is an Fσ set. Furthermore any countable set in a T1 space, is an Fσ set, because a singleton set $\{x\}$  is closed.

The set $\mathbb {R} \setminus \mathbb {Q}$  of irrationals is not a Fσ set.

In metrizable spaces, every open set is an Fσ set.

The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.

The set $A$  of all points $(x,y)$  in the Cartesian plane such that $x/y$  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:

$A=\bigcup _{r\in \mathbb {Q} }\{(ry,y)\mid y\in \mathbb {R} \},$

where $\mathbb {Q}$ , is the set of rational numbers, which is a countable set.