F-sigma set

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

In metrizable spaces, every open set is an F_σ set. The complement of an F_σ set is a G_δ set. In a metrizable space, any closed set is a G_δ set.

The union of countably many F_σ sets is an F_σ set, and the intersection of finitely many F_σ sets is an F_σ set. F_σ is the same as ${\displaystyle \mathbf {\Sigma } _{2}^{0}}$ in the Borel hierarchy.

Each closed set is an F_σ set.

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

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