In general topology, an Fσ set (pronounced 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 Fσ set is a Gδ set.

Fσ in the Borel hierarchy.

Examples

Each closed set is an Fσ set.

of rationalss is an Fσ. More generally, any countable set in a T1 space is an Fσ is closed.

of irrationals is not an Fσ set.

In metrizable spaces, every open set is an Fσ set.

The intersection or union of finitely many Fσ sets is an Fσ set.

Assuming the Axiom of countable choice, the union of countably many Fσ sets is an Fσ set.

of all points in the Cartesian plane is rational is an Fσ set because it can be expressed as the union of all the liness passing through the origin with rational slope:

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

See also

• Gδ set — the dual notion.

• Borel hierarchy

• P-space, any space having the property that every Fσ set is closed

References