Fσ set explained

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

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

F_σ is the same as

	0
\Sigma
	2

in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

The set

of rationals is an F_σ set in

. More generally, any countable set in a T₁ space is an F_σ set, because every singleton

\{x\}

is closed.

The set

R\setminusQ

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

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=cup_r\{(ry,y)\midy\inR\},

where

is the set of rational numbers, which is a countable set.

Fσ set explained

Examples

See also

Notes and References