Fσ 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 fermé (French: closed) and σ for somme (French: sum, union).^[1]

In metrizable spaces, every open set is an F_σ set.^[2] The complement of an F_σ set is a G_δ set.^[1] 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 $\mathbf{\Sigma}^0_2$ in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

In a Tychonoff space, each countable set is an F_σ set, because a point ${x}$ is closed.

For example, 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

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.