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 (''French'': closed) and σ for (''French'': sum, union)..

The complement of an F_σ set is a G_δ set.

F_σ is the same as $\mathbf^0_2$ in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

The set $\mathbb\setminus\mathbb$ of irrationals is not an F_σ set.

In metrizable spaces, every open set is an F_σ 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.

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:

:$A = \bigcup_ \,$

where $\mathbb$ 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

Topology
Descriptive set theory

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