State transition system

From formulasearchengine
Revision as of 12:50, 27 September 2013 by 174.118.114.80 (talk) (Comparison with abstract rewriting systems)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

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).

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. Fσ is the same as Σ20 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 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 Fσ set because it can be expressed as the union of all the lines passing through the origin with rational slope:

A=r{(ry,y)y},

where , is the set of rational numbers, which is a countable set.

See also


Template:Topology-stub