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

*fermé*(

*French*: closed) and σ for

*somme*(

*French*: sum, union).

^{[1]}

The complement of an F_{σ} set is a G_{δ} set.^{[1]}

F_{σ} is the same as in the Borel hierarchy.

## Examples

Each closed set is an F_{σ} set.

The set of rationals is an F_{σ} set. Furthermore any countable set in a T1 space, is an F_{σ} set, because a singleton set is closed.

The set of irrationals is not a 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 in the Cartesian plane such that 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:

where , 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

- ^
^{a}^{b}Stein, Elias M.; Shakarchi, Rami (2009),*Real Analysis: Measure Theory, Integration, and Hilbert Spaces*, Princeton University Press, p. 23, ISBN 9781400835560. **^**Aliprantis, Charalambos D.; Border, Kim (2006),*Infinite Dimensional Analysis: A Hitchhiker's Guide*, Springer, p. 138, ISBN 9783540295877.

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