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 ${\displaystyle \mathbf {\Sigma } _{2}^{0}}$ in the Borel hierarchy.

Examples

Each closed set is an F? set.

The set ${\displaystyle \mathbb {Q} }$ of rationals is an F? set. The set ${\displaystyle \mathbb {R} \setminus \mathbb {Q} }$ of irrationals is not a F? set.

In a Tychonoff space, each countable set is an F? set, because a point ${\displaystyle {x}}$ is closed.

For example, the set ${\displaystyle A}$ of all points ${\displaystyle (x,y)}$ in the Cartesian plane such that ${\displaystyle x/y}$ 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:

${\displaystyle A=\bigcup _{r\in \mathbb {Q} }\{(ry,y)\mid y\in \mathbb {R} \},}$

where ${\displaystyle \mathbb {Q} }$, is the set of rational numbers, which is a countable set.

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.