Fσ 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 Template:Wikt-lang (French: closed) and σ for Template:Wikt-lang (French: sum, union).<ref name="ramtihs">Template:Citation.</ref>

The complement of an F_σ set is a G_δ set.<ref name="ramtihs"/>

F_σ is the same as <math>\mathbf{\Sigma}^0_2</math> in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

The set <math>\mathbb{R}\setminus\mathbb{Q}</math> of irrationals is not an F_σ set.

In metrizable spaces, every open set is an F_σ set.<ref>Template:Citation.</ref>

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 <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_σ 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_δ set — the dual notion.
• Borel hierarchy
• P-space, any space having the property that every F_σ set is closed

References

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