# 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 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 $\mathbf{\Sigma}^0_2$ 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.