Mixmaster universe - Revision history

131.130.45.149: /* Discussion */ more direct link

2014-06-23T10:30:38Z

Discussion: more direct link

← Older revision		Revision as of 12:30, 23 June 2014
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	~~In [[mathematics]], more specifically [[point-set topology]], a '''Moore space''' is a [[developable space\|developable]] [[regular Hausdorff space]]~~. ~~Equivalently~~, ~~a [[topological space]] ''X''~~ is ~~a Moore space if the following conditions hold:~~		Hi there. Let me begin by introducing the writer, her name is Sophia Boon but she by no means truly liked that title. North Carolina is exactly where we've been living for many years and will never move. The preferred pastime for him and his kids is to perform lacross and he'll be starting something else along with it. Since I was 18 I've been working as a bookkeeper but love [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies real psychics] ([http://Www.chk.woobi.co.kr/xe/?document_srl=346069 http://Www.chk.woobi.co.kr/xe/?document_srl=346069]) quickly my wife and I will start our personal business.<br><br>My weblog - free tarot readings ([http://conniecolin.com/xe/community/24580 conniecolin.com])
	* Any two distinct points can be [[separated by ~~neighbourhoods]], and any [[closed set]] and any point in its [[Complement (set theory)\|complement]] can be separated by neighbourhoods~~. ~~(''X''~~ is ~~a [[regular Hausdorff space]].)~~
	* There is a [[countable set\|countable]] collection of [[open cover]]s of '~~'X'', such that~~ for ~~any closed set ''C''~~ and ~~any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is [[disjoint sets\|disjoint]] from ''C''~~. ~~(''X''~~ is ~~a [[developable space]].)~~

	~~Moore spaces are generally interesting in mathematics because they may be applied~~ to ~~prove interesting [[metrization theorem]]s. The concept of a Moore space was formulated by [[R. L. Moore]] in the earlier part of the 20th century.~~

	~~==Examples~~ and ~~properties==~~
	~~#Every [[metrizable space]],~~ '~~'X'', is a Moore space~~. ~~If {~~'~~'A''<sup>(''n'')</sup><sub>''x''</sub>} is the open cover of ''X'' (indexed by ''x'' in ''X'') by all balls of radius 1/''n'', then the collection of all such open covers~~ as ~~''n'' varies over the positive integers is~~ a ~~development of ''X''. Since all metrizable spaces are normal, all metric spaces are Moore spaces.~~
	~~#Moore spaces are a lot like regular spaces and different to~~ [~~[normal space]]s in the sense that every subspace of a Moore space is also a Moore space~~.
	~~#The image of a Moore space under an injective, continuous open map is always a Moore space~~. ~~Note also that the image of a regular space under an injective, continuous open map is always regular.~~
	~~#Both examples 2~~ and ~~3 suggest that Moore spaces are a lot similar to regular spaces.~~
	~~#Neither the [[Sorgenfrey line]] nor the [[Sorgenfrey plane]] are Moore spaces because they are normal and not second countable.~~
	~~#The [[Moore plane]~~] (~~also known as the Niemytski space) is an example of a non-metrizable Moore space.~~
	~~#Every~~ [~~[metacompact]], [[separable space\|separable]], normal Moore space is metrizable~~. ~~This theorem is known as Traylor’s theorem~~.
	~~#Every [[Locally compact space\|locally compact]], [[Locally connected\|locally connected space]], normal Moore space is metrizable~~. ~~This theorem was proved by Reed and Zenor~~.
	~~#If <math>2^{\aleph_0}<2^{\aleph_1}<~~/~~math>, then every [[separable space\|separable]] [[normal space\|normal]] Moore space is [[metrizable]]. This theorem is known as Jones’ theorem.~~

	~~==Normal Moore space conjecture=~~=
	~~For a long time, topologists were trying to prove the so-called normal Moore space conjecture~~: ~~every normal Moore space is [[metrizable]]~~. ~~This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal~~. ~~This would have been a nice [[Metrization theorems\|metrization theorem]]~~. ~~There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section~~.

	~~Here we see that we drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is [[Fleissner's theorem]] that the [[V~~=~~L\|axiom of constructibility~~]~~] implies that locally compact, normal Moore spaces are metrizable.~~

	~~On the other hand, under the [[Continuum hypothesis]] (CH~~) and ~~also under [[Martin's Axiom]] and not CH, there are several examples of non-metrizable normal Moore spaces~~. ~~Nyikos proved that, under the so~~-~~called PMEA~~ (~~Product Measure Extension Axiom), which needs a~~ [[Large cardinal axiom\|large cardinal]], all normal Moore spaces are metrizable. Finally, it was shown later that any model of ZFC in which the conjecture holds, implies the existence of a model with a large cardinal. So large cardinals are needed essentially.

	~~[[Robert Lee Moore\|Moore]] himself proved the theorem that a [[collectionwise normal]] Moore space is metrizable, so strengthening normality is another way to settle the matter.~~

	~~==References==~~
	* [[Lynn Steen\|Lynn Arthur Steen]] and J. Arthur Seebach, ''Counterexamples in Topology'', Dover Books, 1995. ISBN 0-486-68735-X
	* {{PlanetMath attribution\|id=6496\|title=Moore space}}
	* ''The original definition by [[R.L. Moore]] appears here'':
	~~:: MR0150722 (27 #709) Moore, R~~. L. ''Foundations of point set theory''. Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII American Mathematical Society, Providence, R.I. 1962 xi+419 pp. (Reviewer: F. Burton Jones)
	* ''Historical information can be found here'':
	~~:: MR0199840 (33 #7980) Jones, F. Burton "Metrization". ''[[American Mathematical Monthly~~]~~]'' 73 1966 571–576. (Reviewer: R. W. Bagley~~)
	* ''Historical information can be found here'':
	~~:: MR0203661 (34 #3510) Bing, R. H. "Challenging conjectures". ''American Mathematical Monthly'' 74 1967 no. 1, part II, 56–64;~~
	* ''Vickery's theorem may be found here'':
	~~:: MR0001909 (1,317f) Vickery, C. W. "Axioms for Moore spaces and metric spaces". ''Bulletin of the American Mathematical Society'' 46, (1940). 560–564~~

	~~[[Category:General topology]]~~

en>Dirac1234: /* Metric */

2013-04-10T06:31:18Z

Metric

← Older revision		Revision as of 08:31, 10 April 2013
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	~~Andera~~ is ~~what you~~ can ~~call her but she~~ by ~~no means truly liked~~ that ~~title~~. ~~My spouse~~ and ~~I live~~ in ~~Kentucky~~. ~~Distributing production~~ is ~~where her primary income arrives from~~. ~~Playing badminton~~ is a ~~factor~~ that he is ~~completely addicted~~ to.<br><br>~~my web site~~ :~~: cheap psychic readings (~~[~~http://Www~~.~~Prayerarmor~~.~~com/uncategorized/dont~~-~~know~~-which~~-kind-~~of~~-hobby-~~to-~~take~~-up-~~read-the-following-tips/ http~~:~~//Www~~.~~Prayerarmor~~.~~com~~])		In [[mathematics]], more specifically [[point-set topology]], a '''Moore space''' is a [[developable space\|developable]] [[regular Hausdorff space]]. Equivalently, a [[topological space]] ''X'' is a Moore space if the following conditions hold:
			* Any two distinct points can be [[separated by neighbourhoods]], and any [[closed set]] and any point in its [[Complement (set theory)\|complement]] can be separated by neighbourhoods. (''X'' is a [[regular Hausdorff space]].)
			* There is a [[countable set\|countable]] collection of [[open cover]]s of ''X'', such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is [[disjoint sets\|disjoint]] from ''C''. (''X'' is a [[developable space]].)

			Moore spaces are generally interesting in mathematics because they may be applied to prove interesting [[metrization theorem]]s. The concept of a Moore space was formulated by [[R. L. Moore]] in the earlier part of the 20th century.

			==Examples and properties==
			#Every [[metrizable space]], ''X'', is a Moore space. If {''A''<sup>(''n'')</sup><sub>''x''</sub>} is the open cover of ''X'' (indexed by ''x'' in ''X'') by all balls of radius 1/''n'', then the collection of all such open covers as ''n'' varies over the positive integers is a development of ''X''. Since all metrizable spaces are normal, all metric spaces are Moore spaces.
			#Moore spaces are a lot like regular spaces and different to [[normal space]]s in the sense that every subspace of a Moore space is also a Moore space.
			#The image of a Moore space under an injective, continuous open map is always a Moore space. Note also that the image of a regular space under an injective, continuous open map is always regular.
			#Both examples 2 and 3 suggest that Moore spaces are a lot similar to regular spaces.
			#Neither the [[Sorgenfrey line]] nor the [[Sorgenfrey plane]] are Moore spaces because they are normal and not second countable.
			#The [[Moore plane]] (also known as the Niemytski space) is an example of a non-metrizable Moore space.
			#Every [[metacompact]], [[separable space\|separable]], normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
			#Every [[Locally compact space\|locally compact]], [[Locally connected\|locally connected space]], normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
			#If <math>2^{\aleph_0}<2^{\aleph_1}</math>, then every [[separable space\|separable]] [[normal space\|normal]] Moore space is [[metrizable]]. This theorem is known as Jones’ theorem.

			==Normal Moore space conjecture==
			For a long time, topologists were trying to prove the so-called normal Moore space conjecture: every normal Moore space is [[metrizable]]. This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal. This would have been a nice [[Metrization theorems\|metrization theorem]]. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.

			Here we see that we drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is [[Fleissner's theorem]] that the [[V=L\|axiom of constructibility]] implies that locally compact, normal Moore spaces are metrizable.

			On the other hand, under the [[Continuum hypothesis]] (CH) and also under [[Martin's Axiom]] and not CH, there are several examples of non-metrizable normal Moore spaces. Nyikos proved that, under the so-called PMEA (Product Measure Extension Axiom), which needs a [[Large cardinal axiom\|large cardinal]], all normal Moore spaces are metrizable. Finally, it was shown later that any model of ZFC in which the conjecture holds, implies the existence of a model with a large cardinal. So large cardinals are needed essentially.

			[[Robert Lee Moore\|Moore]] himself proved the theorem that a [[collectionwise normal]] Moore space is metrizable, so strengthening normality is another way to settle the matter.

			==References==
			* [[Lynn Steen\|Lynn Arthur Steen]] and J. Arthur Seebach, ''Counterexamples in Topology'', Dover Books, 1995. ISBN 0-486-68735-X
			* {{PlanetMath attribution\|id=6496\|title=Moore space}}
			* ''The original definition by [[R.L. Moore]] appears here'':
			:: MR0150722 (27 #709) Moore, R. L. ''Foundations of point set theory''. Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII American Mathematical Society, Providence, R.I. 1962 xi+419 pp. (Reviewer: F. Burton Jones)
			* ''Historical information can be found here'':
			:: MR0199840 (33 #7980) Jones, F. Burton "Metrization". ''[[American Mathematical Monthly]]'' 73 1966 571–576. (Reviewer: R. W. Bagley)
			* ''Historical information can be found here'':
			:: MR0203661 (34 #3510) Bing, R. H. "Challenging conjectures". ''American Mathematical Monthly'' 74 1967 no. 1, part II, 56–64;
			* ''Vickery's theorem may be found here'':
			:: MR0001909 (1,317f) Vickery, C. W. "Axioms for Moore spaces and metric spaces". ''Bulletin of the American Mathematical Society'' 46, (1940). 560–564

			[[Category:General topology]]

en>Music Sorter: Fixed DAB link

2011-06-27T04:53:14Z

Fixed DAB link

New page

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