- Examples
- See also
- References
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.
Examples
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 also
- Gδ set — the dual notion.
- Borel hierarchy
- P-space, any space having the property that every Fσ set is closed
References
2. ^{{citation|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|first1=Charalambos D.|last1=Aliprantis|first2=Kim|last2=Border|publisher=Springer|year=2006|isbn=9783540295877|page=138|url=https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA138}}.
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