Fσ set

In mathematics, an F_σ set (pronounced 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 F_σ set is a G_δ 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 $\mathbb{Q}$ of rationals is an F_σ set in $ℝ$. More generally, any countable set in a T₁ space is an F_σ set, because every singleton $\{x\}$ is closed.

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

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

The intersection of finitely many F_σ sets is an F_σ set.

Assuming the Axiom of countable choice, the union of countably many F_σ sets is an F_σ set.

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:

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

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

See also

• G_δ set — the dual notion.
• Borel hierarchy
• P-space, any space having the property that every F_σ set is closed