Fσ set
In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union).[1]
The complement of an Fσ set is a Gδ set.[1]
Fσ is the same as [math]\displaystyle{ \mathbf{\Sigma}^0_2 }[/math] in the Borel hierarchy.
Examples
Each closed set is an Fσ set.
The set [math]\displaystyle{ \mathbb{Q} }[/math] of rationals is an Fσ set in [math]\displaystyle{ \mathbb{R} }[/math]. More generally, any countable set in a T1 space is an Fσ set, because every singleton [math]\displaystyle{ \{x\} }[/math] is closed.
The set [math]\displaystyle{ \mathbb{R}\setminus\mathbb{Q} }[/math] of irrationals is not an Fσ set.
In metrizable spaces, every open set is an Fσ set.[2]
The union of countably many Fσ sets is an Fσ set, and the intersection of finitely many Fσ sets is an Fσ set.
The set [math]\displaystyle{ A }[/math] of all points [math]\displaystyle{ (x,y) }[/math] in the Cartesian plane such that [math]\displaystyle{ x/y }[/math] 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:
- [math]\displaystyle{ A = \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\}, }[/math]
where [math]\displaystyle{ \mathbb{Q} }[/math] 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
- ↑ 1.0 1.1 Stein, Elias M.; Shakarchi, Rami (2009), Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press, p. 23, ISBN 9781400835560, https://books.google.com/books?id=2Sg3Vug65AsC&pg=PA23.
- ↑ Aliprantis, Charalambos D.; Border, Kim (2006), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 138, ISBN 9783540295877, https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA138.
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