In mathematics, an F<sub>σ</sub> set (pronounced 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 F<sub>σ</sub> set is a G<sub>δ</sub> set.
The intersection or union of finitely many F<sub>σ</sub> sets is an F<sub>σ</sub> set.
Assuming the Axiom of countable choice, the union of countably many F<sub>σ</sub> sets is an F<sub>σ</sub> set.
The set <math>A</math> of all points <math>(x,y)</math> in the Cartesian plane such that <math>x/y</math> is rational is an F<sub>σ</sub> set because it can be expressed as the union of all the lines passing through the origin with rational slope:
:<math> A = \bigcup_{r \in \mathbb{Q \{(ry,y) \mid y \in \mathbb{R}\},</math>
where <math>\mathbb{Q}</math> is the set of rational numbers, which is a countable set.
See also
- G<sub>δ</sub> set — the dual notion.
- Borel hierarchy
- P-space, any space having the property that every F<sub>σ</sub> set is closed