Fσ set

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for (French: closed) and σ for  (French: sum, union).

The complement of an Fσ set is a G&delta; 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 in $$\mathbb{R}$$. More generally, any countable set in a T1 space is an Fσ set, because every singleton $$\{x\}$$ is closed.

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

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

The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.

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.