F-sigma Set
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F-sigma Set

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

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

The union of countably many F? sets is an F? set, and the intersection of finitely many F? sets is an F? set. F? is the same as in the Borel hierarchy.

Examples

Each closed set is an F? set.

The set of rationals is an F? set. The set of irrationals is not a F? set.

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

For example, the set of all points in the Cartesian plane such that is rational is an F? 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

References

  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.



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