174.118.114.80: /* Comparison with abstract rewriting systems */

2013-09-27T11:50:04Z

Comparison with abstract rewriting systems

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			In mathematics, an '''F<sub>σ</sub> set''' (said '''F-sigma set''') is a [[countable]] [[union (set theory)\|union]] of [[closed set]]s. The notation originated in [[France]] with F for ''[[wikt:fermé#French\|fermé]]'' (''[[French language\|French]]'': closed) and σ for ''[[wikt:somme#French\|somme]]'' (''French'': sum, union).

			In [[metrizable]] spaces, every [[open set]] is an F<sub>σ</sub> set.
			The complement of an F<sub>σ</sub> set is a [[G-delta set\|G<sub>δ</sub> set]]. In a metrizable space, any closed set is a G<sub>δ</sub> set.

			The union of countably many F<sub>σ</sub> sets is an F<sub>σ</sub> set, and the intersection of finitely many F<sub>σ</sub> sets is an F<sub>σ</sub> set. F<sub>σ</sub> is the same as <math>\mathbf{\Sigma}^0_2</math> in the Borel hierarchy.

			== Examples ==

			Each closed set is an F<sub>σ</sub> set.

			The set <math>\mathbb{Q}</math> of rationals is an F<sub>σ</sub> set. The set <math>\mathbb{R}\setminus\mathbb{Q}</math> of irrationals is not a F<sub>σ</sub> set.

			In a Tychonoff space, each countable set is an F<sub>σ</sub> set, because a point <math>{x}</math> is closed.

			For example, the set <math>A</math> of all [[Point (geometry)\|point]]s <math>(x,y)</math> in the [[Cartesian plane]] such that <math>x/y</math> is [[rational number\|rational]] is an F<sub>σ</sub> set because it can be expressed as the union of all the [[line (mathematics)\|line]]s passing through the [[origin (mathematics)\|origin]] with rational [[slope (mathematics)\|slope]]:

			:<math> A = \bigcup_{r \in \mathbb{Q}} \{(ry,y) \mid y \in \mathbb{R}\},</math>

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

			==See also==
			*[[G-delta set\|G<sub>δ</sub> set]] — the [[duality (mathematics)\|dual]] notion.
			*[[Borel hierarchy]]
			* [[P-space\|''P''-space]], any space having the property that every F<sub>σ</sub> set is closed

			{{DEFAULTSORT:F Set}}
			[[Category:Topology]]
			[[Category:Descriptive set theory]]


			{{topology-stub}}

en>Dougher: /* Comparison with abstract rewriting systems */ spell out mnemonic

2012-06-16T02:48:16Z

Comparison with abstract rewriting systems: spell out mnemonic

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174.118.114.80: /* Comparison with abstract rewriting systems */

en>Dougher: /* Comparison with abstract rewriting systems */ spell out mnemonic