# 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 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 $\mathbf {\Sigma } _{2}^{0}$ in the Borel hierarchy.

## Examples

Each closed set is an Fσ set.

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

In a Tychonoff space, each countable set is an Fσ set, because a point ${x}$  is closed.

For example, 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.