# Fσ set

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# Fσ set

Fσ set

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.com

*Real Analysis: Measure Theory, Integration, and Hilbert Spaces*Sep 26, 2019, 12:44 AM

[4]

Citation Linkbooks.google.com

*Infinite 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