en>Rjwilmsi: /* Interdisciplinary */Journal cites, Added 1 doi to a journal cite using AWB (10473)

2014-09-25T12:32:14Z

Interdisciplinary: Journal cites, Added 1 doi to a journal cite using AWB (10473)

18.82.8.137: added missing period

2013-11-10T20:25:59Z

added missing period

← Older revision		Revision as of 22:25, 10 November 2013
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	~~Hello! My name~~ is ~~Beatriz.~~ <br>It is a ~~little about myself~~: ~~I live~~ in ~~United States~~, ~~my city of San Jose.~~ <br>It'~~s called often Eastern~~ or ~~cultural capital~~ of CA. I'~~ve married 1 years ago~~.<br>~~I have two children -~~ a ~~son~~ (~~Julia~~) ~~and~~ the ~~daughter (Edwina)~~. ~~We all like Card collecting~~.<br><br>~~Also visit my webpage fifa 15 coin generator (~~[~~http:~~//~~web38~~.~~wwwsrv03~~.~~t3-Network~~.~~de/guestbook/guestbook~~.~~php http~~:~~//web38~~.~~wwwsrv03~~.t3-~~network.de~~])		In [[set theory]], the '''axiom of uniformization''', a weak form of the [[axiom of choice]], states that if <math>R</math> is a [[subset]] of <math>X\times Y</math>, where <math>X</math> and <math>Y</math> are [[Polish space]]s,
			then there is a subset <math>f</math> of <math>R</math> that is a [[partial function]] from <math>X</math> to <math>Y</math>, and whose domain (in the sense of the set of all <math>x</math> such that <math>f(x)</math> exists) equals
			: <math>\{x\in X\|\exists y\in Y (x,y)\in R\}\,</math>
			Such a function is called a '''uniformizing function''' for <math>R</math>, or a '''uniformization''' of <math>R</math>.

			[[Image:Uniformization ill.png\|thumb\|right\|Uniformization of relation ''R'' (light blue) by function ''f'' (red).]]

			To see the relationship with the axiom of choice, observe that <math>R</math> can be thought of as associating, to each element of <math>X</math>, a subset of <math>Y</math>. A uniformization of <math>R</math> then picks exactly one element from each such subset, whenever the subset is [[nonempty]]. Thus, allowing arbitrary sets ''X'' and ''Y'' (rather than just Polish spaces) would make the axiom of uniformization equivalent to AC.

			A [[pointclass]] <math>\boldsymbol{\Gamma}</math> is said to have the '''uniformization property''' if every relation <math>R</math> in <math>\boldsymbol{\Gamma}</math> can be uniformized by a partial function in <math>\boldsymbol{\Gamma}</math>. The uniformization property is implied by the [[scale property]], at least for [[adequate pointclass]]es of a certain form.

			It follows from [[ZFC]] alone that <math>\boldsymbol{\Pi}^1_1</math> and <math>\boldsymbol{\Sigma}^1_2</math> have the uniformization property. It follows from the existence of sufficient [[large cardinal]]s that
			*<math>\boldsymbol{\Pi}^1_{2n+1}</math> and <math>\boldsymbol{\Sigma}^1_{2n+2}</math> have the uniformization property for every [[natural number]] <math>n</math>.
			*Therefore, the collection of [[projective set]]s has the uniformization property.
			*Every relation in [[L(R)]] can be uniformized, but ''not necessarily'' by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization).
			**(Note: it's trivial that every relation in L(R) can be uniformized ''in V'', assuming V satisfies AC. The point is that every such relation can be uniformized in some transitive inner model of V in which AD holds.)

			== References ==

			* {{cite book \| author=Moschovakis, Yiannis N. \| title=Descriptive Set Theory \| publisher=North Holland \| year=1980 \|isbn=0-444-70199-0}}

			[[Category:Set theory]]
			[[Category:Descriptive set theory]]
			[[Category:Axiom of choice]]

en>Chris the speller: /* Spatial Autocorrelation */Typo fixing, replaced: over-estimate → overestimate using AWB (8207)

2012-08-04T21:31:29Z

Spatial Autocorrelation: Typo fixing, replaced: over-estimate → overestimate using AWB (8207)

New page

Hello! My name is Beatriz. It is a little about myself: I live in United States, my city of San Jose. It's called often Eastern or cultural capital of CA. I've married 1 years ago. I have two children - a son (Julia) and the daughter (Edwina). We all like Card collecting. Also visit my webpage fifa 15 coin generator ([http://web38.wwwsrv03.t3-Network.de/guestbook/guestbook.php http://web38.wwwsrv03.t3-network.de])

Spatial ecology - Revision history

en>Rjwilmsi: /* Interdisciplinary */Journal cites, Added 1 doi to a journal cite using AWB (10473)

18.82.8.137: added missing period

en>Chris the speller: /* Spatial Autocorrelation */Typo fixing, replaced: over-estimate → overestimate using AWB (8207)