Topological defect - Revision history

en>Omnipaedista: per MOS:AMP

2014-12-21T17:22:22Z

← Older revision		Revision as of 19:22, 21 December 2014
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en>Primergrey: /* Condensed matter */

2014-03-01T13:22:50Z

Condensed matter

← Older revision		Revision as of 15:22, 1 March 2014
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	~~In [[mathematics]], a '''Cantor space''', named for [[Georg Cantor]],~~ is ~~a [[topology\|topological]] abstraction of the classical [[Cantor set]]: a [[topological space]] is a~~ ''~~'Cantor space''' if it is [[homeomorphic]] to the [[Cantor set]]. In [[set theory]], the topological space 2<sup>ω</sup> is~~ called ~~"the" Cantor space~~. ~~Note that, commonly, 2<sup>ω</sup> is referred~~ to ~~simply as the Cantor set, while the term Cantor space~~ is ~~reserved for the more general construction of ''D''<sup>S</sup> for a finite set~~ '~~'D'' and a set ''S'' which might be finite, countable or possibly uncountable.<ref>Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing. ''See section 17.9a''</ref>~~		Gabrielle is what her his conversation loves to call lady's though she doesn't totally like being called like this. Fish hanging on to is something her husband doesn't really like but she does. Managing people is [http://www.britannica.com/search?query=regarded regarded] as what she does on the other hand she plans on diaper changing it. For years she's been daily living in [http://Thesaurus.com/browse/Massachusetts Massachusetts]. Go to her website to find to choose from more: http://circuspartypanama.com<br><br>Stop by my site :: [http://circuspartypanama.com clash of clans hack]

	~~== Examples ==~~

	~~The Cantor set itself is a Cantor space~~. ~~But the canonical example of a Cantor space~~ is ~~the~~ [~~[countably infinite]] [[product topology\|topological product]] of the [[discrete 2-point space]] {0, 1}. This is usually written as <math>2^\mathbb{N}<~~/~~math> or 2<sup>ω<~~/~~sup> (where 2 denotes the 2-element set {0,1} with the discrete topology)~~. ~~A point in 2<sup>ω</sup> is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1~~. ~~Given such a sequence ''a''<sub>0</sub>, ''a''<sub>1<~~/~~sub>, ''a''<sub>2</sub>,..., one can map it to the real number~~
	~~:<math>~~
	~~\sum_{n~~=~~0}^\infty \frac{2 a_n}{3^{n+1}}.~~
	~~</math>~~
	~~This mapping gives a homeomorphism from 2<sup>ω</sup> onto the [[Cantor set~~]~~], demonstrating that 2<sup>ω</sup> is indeed a Cantor space.~~

	~~Cantor spaces occur abundantly in [[real analysis]]. For example, they exist~~ as subspaces in every perfect, [[complete space\|complete]] [[metric space]]. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the ~~construction~~
	~~of the usual [[Cantor set]].) Also, every uncountable,~~
	~~[[separable space\|separable]], completely metrizable space contains~~
	~~Cantor spaces as subspaces. This includes most of the common type of spaces in real analysis.~~

	~~== Characterization ==~~

	~~A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer\|Brouwer]]'s theorem:{{citation needed\|date=February 2014}}~~
	~~:''Any two non-empty [[compact set\|compact]] [[Hausdorff space]]s without [[isolated point]]s and having countable [[base (topology)\|base]]s consisting of [[clopen set]]s are homeomorphic to each~~ other~~''.~~
	~~The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:~~
	~~:''A topological space is a Cantor space if and only if~~ it ~~is non-empty, [[perfect space\|perfect]], [[compact space\|compact]], [[totally disconnected]], and [[metrizable]]~~.''
	~~This theorem is also equivalent (via [[Stone~~'s ~~representation theorem for Boolean algebras]]) to the fact that any two countable atomless~~ [~~[Boolean algebra (structure)\|Boolean algebra]]s are isomorphic~~.

	~~== Properties ==~~
	~~As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2<sup>ω<~~/~~sup>, because its construction as a product makes it amenable to analysis.~~

	~~Cantor spaces have the following properties:~~
	* The [[cardinality]] of any Cantor space is <math>2^{\aleph_0}</~~math>, that is, the [[cardinality of the continuum]~~].
	* The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the [[Cantor function]]; this fact can be used to ~~construct [[space-filling curve]]s.~~
	* A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', London Mathematical Society Student Texts '''64''', (2005) Cambridge University Press. ''See Chapter 12''</~~ref><ref>Willard, ''op.cit.'', ''See section 30.7''<~~/~~ref>~~

	Let ''C(X)'' denote the space of all real-valued, bounded continuous functions on a topological space ''X''. Let ''K'' denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:
	* C(K) is [[isometry\|isometric]] to a closed subspace of C(Δ).<~~ref~~>~~Carothers, ''op.cit.''~~<~~/ref~~>
	~~In general, this isometry is not unique, and thus is not properly a~~ [~~[universal property]] in the categorical sense.~~

	*The group of all [[homeomorphisms]] of the Cantor space is [[simple group\|simple]].<ref>R.D. Anderson, ''The Algebraic Simplicity of Certain Groups of Homeomorphisms'', American Journal of Mathematics '''80''' (1958), pp. 955-963.</~~ref>~~

	~~==See also==~~
	*[[Cantor cube]]
	*[[Georg Cantor]]

	~~==References==~~
	~~<references~~/>
	*{{cite book \| author=Kechris, A. ~~\| title= Classical Descriptive Set Theory \| publisher=Springer \| year=1995 \| isbn = 0-387-94374-9\| edition=Graduate Texts in Mathematics 156}}~~

	~~{{DEFAULTSORT:Cantor Space}}~~
	~~[[Category:Topological spaces]]~~
	~~[[Category:Descriptive set theory]~~]

en>Xiaoshan Math at 18:16, 22 January 2014

2014-01-22T18:16:26Z

← Older revision		Revision as of 20:16, 22 January 2014
Line 1:		Line 1:
	~~Roberto~~ is ~~what~~'~~s written located on his birth certificate fortunately he never really enjoyed reading~~ that ~~name~~. ~~Managing people~~ is ~~undoubtedly~~ where ~~his primary financial comes from~~. ~~Base jumping~~ is ~~something~~ that ~~they have been doing for months~~. ~~[http~~://~~Massachusetts~~.~~com/ Massachusetts~~] ~~has always been his everyday life place~~ and ~~his family loves~~ it. Go of ~~his website~~ to ~~come out more: http~~://~~prometeu~~.~~net~~<br><br>~~Here~~ is ~~my page~~ :: [~~http~~:~~//prometeu.net triche clash of clans~~]		In [[mathematics]], a '''Cantor space''', named for [[Georg Cantor]], is a [[topology\|topological]] abstraction of the classical [[Cantor set]]: a [[topological space]] is a '''Cantor space''' if it is [[homeomorphic]] to the [[Cantor set]]. In [[set theory]], the topological space 2<sup>ω</sup> is called "the" Cantor space. Note that, commonly, 2<sup>ω</sup> is referred to simply as the Cantor set, while the term Cantor space is reserved for the more general construction of ''D''<sup>S</sup> for a finite set ''D'' and a set ''S'' which might be finite, countable or possibly uncountable.<ref>Stephen Willard, ''General Topology'' (1970) Addison-Wesley Publishing. ''See section 17.9a''</ref>

			== Examples ==

			The Cantor set itself is a Cantor space. But the canonical example of a Cantor space is the [[countably infinite]] [[product topology\|topological product]] of the [[discrete 2-point space]] {0, 1}. This is usually written as <math>2^\mathbb{N}</math> or 2<sup>ω</sup> (where 2 denotes the 2-element set {0,1} with the discrete topology). A point in 2<sup>ω</sup> is an infinite binary sequence, that is a sequence which assumes only the values 0 or 1. Given such a sequence ''a''<sub>0</sub>, ''a''<sub>1</sub>, ''a''<sub>2</sub>,..., one can map it to the real number
			:<math>
			\sum_{n=0}^\infty \frac{2 a_n}{3^{n+1}}.
			</math>
			This mapping gives a homeomorphism from 2<sup>ω</sup> onto the [[Cantor set]], demonstrating that 2<sup>ω</sup> is indeed a Cantor space.

			Cantor spaces occur abundantly in [[real analysis]]. For example, they exist as subspaces in every perfect, [[complete space\|complete]] [[metric space]]. (To see this, note that in such a space, any non-empty perfect set contains two disjoint non-empty perfect subsets of arbitrarily small diameter, and so one can imitate the construction
			of the usual [[Cantor set]].) Also, every uncountable,
			[[separable space\|separable]], completely metrizable space contains
			Cantor spaces as subspaces. This includes most of the common type of spaces in real analysis.

			== Characterization ==

			A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer\|Brouwer]]'s theorem:{{citation needed\|date=February 2014}}
			:''Any two non-empty [[compact set\|compact]] [[Hausdorff space]]s without [[isolated point]]s and having countable [[base (topology)\|base]]s consisting of [[clopen set]]s are homeomorphic to each other''.
			The topological property of having a base consisting of clopen sets is sometimes known as "zero-dimensionality". Brouwer's theorem can be restated as:
			:''A topological space is a Cantor space if and only if it is non-empty, [[perfect space\|perfect]], [[compact space\|compact]], [[totally disconnected]], and [[metrizable]].''
			This theorem is also equivalent (via [[Stone's representation theorem for Boolean algebras]]) to the fact that any two countable atomless [[Boolean algebra (structure)\|Boolean algebra]]s are isomorphic.

			== Properties ==
			As can be expected from Brouwer's theorem, Cantor spaces appear in several forms. But many properties of Cantor spaces can be established using 2<sup>ω</sup>, because its construction as a product makes it amenable to analysis.

			Cantor spaces have the following properties:
			* The [[cardinality]] of any Cantor space is <math>2^{\aleph_0}</math>, that is, the [[cardinality of the continuum]].
			* The product of two (or even any finite or countable number of) Cantor spaces is a Cantor space. Along with the [[Cantor function]]; this fact can be used to construct [[space-filling curve]]s.
			* A Hausdorff topological space is compact metrizable if and only if it is a continuous image of a Cantor space.<ref>N.L. Carothers, ''A Short Course on Banach Space Theory'', London Mathematical Society Student Texts '''64''', (2005) Cambridge University Press. ''See Chapter 12''</ref><ref>Willard, ''op.cit.'', ''See section 30.7''</ref>

			Let ''C(X)'' denote the space of all real-valued, bounded continuous functions on a topological space ''X''. Let ''K'' denote a compact metric space, and Δ denote the Cantor set. Then the Cantor set has the following property:
			* C(K) is [[isometry\|isometric]] to a closed subspace of C(Δ).<ref>Carothers, ''op.cit.''</ref>
			In general, this isometry is not unique, and thus is not properly a [[universal property]] in the categorical sense.

			*The group of all [[homeomorphisms]] of the Cantor space is [[simple group\|simple]].<ref>R.D. Anderson, ''The Algebraic Simplicity of Certain Groups of Homeomorphisms'', American Journal of Mathematics '''80''' (1958), pp. 955-963.</ref>

			==See also==
			*[[Cantor cube]]
			*[[Georg Cantor]]

			==References==
			<references/>
			*{{cite book \| author=Kechris, A. \| title= Classical Descriptive Set Theory \| publisher=Springer \| year=1995 \| isbn = 0-387-94374-9\| edition=Graduate Texts in Mathematics 156}}

			{{DEFAULTSORT:Cantor Space}}
			[[Category:Topological spaces]]
			[[Category:Descriptive set theory]]

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2012-08-25T15:57:10Z

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