Fσ set

In general topology, 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 $\mathbf {\Sigma } _{2}^{0}$ 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 $\mathbb {R}$. More generally, any countable set in a T₁ space is an F_σ set, because every singleton $\{x\}$ is closed.

The set $\mathbb {R} \setminus \mathbb {Q}$ 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:

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

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