Fσ set

Short description: Countable union of closed sets

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 ${\mathrm{\Sigma }}_2^0$ in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

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

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

The intersection or union 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:

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

where $\mathbb{\mathbb{Q}}$ 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