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 {{wikt-lang|fr|fermé}} (French: closed) and σ for {{wikt-lang|fr|somme}} (French: sum, union).{{citation|title=Real Analysis: Measure Theory, Integration, and Hilbert Spaces|first1=Elias M.|last1=Stein|author1link = Elias M. Stein|first2=Rami|last2=Shakarchi|publisher=Princeton University Press|year=2009|isbn=9781400835560|page=23|url=https://books.google.com/books?id=2Sg3Vug65AsC&pg=PA23}}.

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

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

In metrizable spaces, every open set is an F_σ set.{{citation|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|first1=Charalambos D.|last1=Aliprantis|author1link = Charalambos D. Aliprantis|first2=Kim|last2=Border|publisher=Springer|year=2006|isbn=9783540295877|page=138|url=https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA138}}.

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 $A$ 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.

References

Category:Topology

Category:Descriptive set theory

Fσ set

Examples

See also

References