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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 in the Borel hierarchy.

ExamplesEdit

Each closed set is an Fσ set.

The set   of rationals is an Fσ set. The set   of irrationals is not a Fσ set.

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

For example, the set   of all points   in the Cartesian plane such that   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:

 

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

See alsoEdit

ReferencesEdit

  1. ^ a b 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.