Fσ set

In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in France with F for fermé (French: closed) and σ for somme (French: sum, union).[1]

In metrizable spaces, every open set is an Fσ set.[2] The complement of an Fσ set is a Gδ set.[1] In a metrizable space, any closed set is a Gδ set.

The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set. Fσ is the same as \mathbf{\Sigma}^0_2 in the Borel hierarchy.

Examples

Each closed set is an Fσ set.

The set \mathbb{Q} of rationals is an Fσ set. The set \mathbb{R}\setminus\mathbb{Q} of irrationals is not a Fσ set.

In a Tychonoff space, each countable set is an Fσ set, because a point {x} is closed.

For example, the set A of all points (x,y) in the Cartesian plane such that x/y is rational is an Fσ 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.

See also

References

  1. ↑ 1.0 1.1 Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.
  2. ↑ Aliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 138, ISBN 9783540295877.