F_σ set

In mathematics, an F_σ set (said 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).^[1]

The complement of an F_σ set is a G_δ set.^[1]

F_σ is the same as ${\displaystyle \mathbf {\Sigma } _{2}^{0}}$ in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

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

In metrizable spaces, every open set is an F_σ set.^[2]

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.

The set ${\displaystyle A}$  of all points ${\displaystyle (x,y)}$  in the Cartesian plane such that ${\displaystyle x/y}$  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:

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

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

References

1. Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.
2. Aliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 138, ISBN 9783540295877.