Fσ set

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The correct title of this article is Fσ set. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

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: union).

In metrizable spaces, every open set is an Fσ set.

The complement of an Fσ set is a Gδ set. 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.

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

A = \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\},

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

[edit] See also

Gδ set — the dual notion.

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