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Fσ set

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 Fσsets is an Fσset, and the intersection of finitely many Fσsets is an Fσset. Fσis the same asin theBorel hierarchy.

Examples

Each closed set is an Fσ set.

The setof rationals is an Fσset. The setof irrationals is not a Fσset.
In a Tychonoff space, each countable set is an Fσset, because a pointis closed.
For example, the setof allpointsin theCartesian planesuch thatisrationalis an Fσset because it can be expressed as the union of all thelinespassing through theoriginwith rationalslope:
where, 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

[1]
Citation Linkbooks.google.comStein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560.
Sep 26, 2019, 12:44 AM
[2]
Citation Linkbooks.google.comAliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 138, ISBN 9783540295877.
Sep 26, 2019, 12:44 AM
[3]
Citation Linkbooks.google.comReal Analysis: Measure Theory, Integration, and Hilbert Spaces
Sep 26, 2019, 12:44 AM
[4]
Citation Linkbooks.google.comInfinite Dimensional Analysis: A Hitchhiker's Guide
Sep 26, 2019, 12:44 AM
[5]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
Sep 26, 2019, 12:44 AM