State transition system

In mathematics, an F_σ 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).

In metrizable spaces, every open set is an F_σ set. The complement of an F_σ set is a G_δ set. In a metrizable space, any closed set is a G_δ set.

The union of countably many F_σ sets is an F_σ set, and the intersection of finitely many F_σ sets is an F_σ set. F_σ is the same as ${\mathrm{\Sigma }}_2^0$ in the Borel hierarchy.

Examples

Each closed set is an F_σ set.

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

In a Tychonoff space, each countable set is an F_σ set, because a point $x$ is closed.

For example, the set $A$ of all points $(x,y)$ in the Cartesian plane such that $x/y$ is rational is an F_σ set because it can be expressed as the union of all the lines passing through the origin with rational slope:

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

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

See also

• G_δ set — the dual notion.
• Borel hierarchy
• P-space, any space having the property that every F_σ set is closed

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