Integral symbol - Revision history

174.3.125.23: unrelated offtopic tanget

2014-08-29T22:37:43Z

unrelated offtopic tanget

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	~~In the mathematical field of~~ [~~[topology]], the '''inductive dimension''' of a [[topological space]] ''X'' is either of two values, the '''small inductive dimension''' ind(''X'') or the '''large inductive dimension''' Ind(''X'')~~. ~~These are based on the observation that, in ''n''-dimensional [[Euclidean space]] ''R''<sup>''n''<~~/sup>, (''n'' − 1)-dimensional [[sphere]]s (that is, the [[boundary (topology)\|boundaries]] of ''n''-dimensional balls) have dimension ''n'' − 1. Therefore it should be possible to define the dimension of a space [[mathematical induction\|inductively]] in terms of the dimensions of the boundaries of suitable [[open set]]s.		The person who wrote [http://ustanford.com/index.php?do=/profile-38218/info/ certified psychics] the post is known as Jayson Hirano and he totally digs that name. My day job is an invoicing officer but I've already applied for another one. [http://www.familysurvivalgroup.com/easy-methods-planting-looking-backyard/ real psychic] The favorite pastime for him and his children is to perform lacross and he would never give it up. Some time in the past he selected to reside in North Carolina and he doesn't plan on altering it.<br><br>Feel free to surf to my homepage; [http://www.article-galaxy.com/profile.php?a=143251 are psychics real]

	The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a [[metric space]]). The other is the [[Lebesgue covering dimension]]. The term "topological dimension" is ordinarily understood to refer to Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.

	~~==Formal definition==~~
	~~We want the dimension of a point to be 0, and a point has empty boundary, so we start with~~

	~~:<math>\operatorname{ind}(\varnothing)=\operatorname{Ind}(\varnothing)~~=-1</~~math>~~

	~~Then inductively, ind(''X'') is the smallest ''n'' such that, for every ''<math>x \isin X<~~/~~math>'' and every open set ''U'' containing ''x'', there is an open ''V'' containing ''x'', where~~ the ~~[[Closure (topology)\|closure]] of ''V''~~ is ~~a [[subset]] of ''U'', such~~ that ~~the boundary of ''V'' has small inductive dimension less than or equal to ''n'' − 1~~. ~~(In the case above, where ''X''~~ is ~~Euclidean ''n''-dimensional space, ''V'' will be chosen to be~~ an '~~'n''-dimensional ball centered at ''x''.)~~

	~~For the large inductive dimension, we restrict the choice of ''V'' still further; Ind(''X'') is the smallest ''n'' such that,~~ for every [[closed set\|closed]] subset ''F'' of every open subset ''U'' of ''X'', there is an open ''V'' in between (that is, ''F'' is a subset of ''V'' and the closure of ''V'' is a subset of ''U''), such that the boundary of ''V'' has large inductive dimension less than or equal to ''n'' − 1.

	~~==Relationship between dimensions==~~

	~~Let <math>\operatorname{dim}</math> be the Lebesgue covering dimension~~. ~~For any [~~[~~topological space]] ''X'', we have~~

	:~~<math>\operatorname{dim} X = 0<~~/~~math> if and only if <math>\operatorname{Ind} X = 0~~.</~~math>~~

	~~'''Urysohn's theorem''' states that when ''X'' is a [[normal space]] with a [[second~~-~~countable space\|countable base]], then~~

	~~:<math>\operatorname{dim} X = \operatorname{Ind} X = \operatorname{ind} X<~~/~~math>.~~

	~~Such spaces are exactly the [[separable space\|separable]~~] and ~~[[metrizable]] ''X'' (see [[Urysohn's metrization theorem]]).~~

	~~The '''Nöbeling-Pontryagin theorem''' then states that such spaces with finite dimension are characterised up~~ to ~~homeomorphism as the subspaces of the [[Euclidean space]]s, with their usual topology. The '''Menger-Nöbeling theorem''' (1932) states that if ''X'' is compact metric separable~~ and ~~of dimension ''n'', then~~ it ~~embeds as a subspace of Euclidean space of dimension 2''n'' + 1~~. ~~([[Georg Nöbeling]] was a student of [[Karl Menger]]. He introduced '''Nöbeling space''',~~ the subspace of '''R'''<sup>2''n'' + 1</sup> consisting of points with at least ''n'' + 1 co-ordinates being [[irrational number]]s, which has universal properties for embedding spaces of dimension ''n''.)

	~~Assuming only ''X'' metrizable we have ([[Miroslav Katětov]])~~

	~~:ind ''X'' ≤ Ind ''X'' = dim ''X'';~~

	~~or assuming ''X'' [[compact space\|compact]]~~ and ~~[[Hausdorff space\|Hausdorff]] ([[P. S. Aleksandrov]])~~

	~~:dim~~ '~~'X'' ≤ ind ''X'' ≤ Ind ''X''.~~

	~~Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ~~.

	~~A separable metric space ''X'' satisfies the inequality~~ <~~math~~>~~\operatorname{Ind}X\le n~~<~~/math~~> ~~if and only if for every closed sub-space <math>A</math> of the space <math>X</math> and each continuous mapping <math>f:A\~~to ~~S^n</math> there exists a continuous extension <math>\bar f:X\~~to ~~S^n</math>.~~

	~~==References==~~
	~~{{Empty section\|date=July 2010}}~~
	~~==Further reading==~~
	*Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in [~~[Ivor Grattan-Guinness\|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier~~: ~~844-55~~.
	*R. Engelking, ''Theory of Dimensions. Finite and Infinite'', Heldermann Verlag (1995), ISBN 3-88538-010-2.
	*V. V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
	*V. V. Filippov, ''On the inductive dimension of the product of bicompacta'', Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.
	*A. R. Pears, ''Dimension theory of general spaces'', Cambridge University Press (1975).

	~~[[Category:Dimension theory]~~]

en>EuroCarGT: Reverted 1 edit by Radcliel identified as test/vandalism using STiki - Mistake? - Tell me here

2013-10-06T03:13:42Z

Reverted 1 edit by Radcliel identified as test/vandalism using STiki - Mistake? - Tell me here

← Older revision		Revision as of 05:13, 6 October 2013
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	~~I would like~~ to ~~introduce myself~~ to ~~you~~, ~~I am Andrew~~ and ~~my spouse doesn~~'~~t like it~~ at ~~all~~. ~~What I love performing~~ is ~~soccer but I don~~'t have the ~~time lately~~. ~~Office supervising~~ is ~~what she does for~~ a ~~residing~~. ~~Her family life in Ohio~~.<br><br>~~my webpage online psychic chat~~; [~~http~~://~~galab~~-~~work~~.cs.~~pusan~~.ac.~~kr/Sol09B/?document_srl=1489804 simply click~~ the ~~following webpage~~],		In the mathematical field of [[topology]], the '''inductive dimension''' of a [[topological space]] ''X'' is either of two values, the '''small inductive dimension''' ind(''X'') or the '''large inductive dimension''' Ind(''X''). These are based on the observation that, in ''n''-dimensional [[Euclidean space]] ''R''<sup>''n''</sup>, (''n'' − 1)-dimensional [[sphere]]s (that is, the [[boundary (topology)\|boundaries]] of ''n''-dimensional balls) have dimension ''n'' − 1. Therefore it should be possible to define the dimension of a space [[mathematical induction\|inductively]] in terms of the dimensions of the boundaries of suitable [[open set]]s.

			The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a [[metric space]]). The other is the [[Lebesgue covering dimension]]. The term "topological dimension" is ordinarily understood to refer to Lebesgue covering dimension. For "sufficiently nice" spaces, the three measures of dimension are equal.

			==Formal definition==
			We want the dimension of a point to be 0, and a point has empty boundary, so we start with

			:<math>\operatorname{ind}(\varnothing)=\operatorname{Ind}(\varnothing)=-1</math>

			Then inductively, ind(''X'') is the smallest ''n'' such that, for every ''<math>x \isin X</math>'' and every open set ''U'' containing ''x'', there is an open ''V'' containing ''x'', where the [[Closure (topology)\|closure]] of ''V'' is a [[subset]] of ''U'', such that the boundary of ''V'' has small inductive dimension less than or equal to ''n'' − 1. (In the case above, where ''X'' is Euclidean ''n''-dimensional space, ''V'' will be chosen to be an ''n''-dimensional ball centered at ''x''.)

			For the large inductive dimension, we restrict the choice of ''V'' still further; Ind(''X'') is the smallest ''n'' such that, for every [[closed set\|closed]] subset ''F'' of every open subset ''U'' of ''X'', there is an open ''V'' in between (that is, ''F'' is a subset of ''V'' and the closure of ''V'' is a subset of ''U''), such that the boundary of ''V'' has large inductive dimension less than or equal to ''n'' − 1.

			==Relationship between dimensions==

			Let <math>\operatorname{dim}</math> be the Lebesgue covering dimension. For any [[topological space]] ''X'', we have

			:<math>\operatorname{dim} X = 0</math> if and only if <math>\operatorname{Ind} X = 0.</math>

			'''Urysohn's theorem''' states that when ''X'' is a [[normal space]] with a [[second-countable space\|countable base]], then

			:<math>\operatorname{dim} X = \operatorname{Ind} X = \operatorname{ind} X</math>.

			Such spaces are exactly the [[separable space\|separable]] and [[metrizable]] ''X'' (see [[Urysohn's metrization theorem]]).

			The '''Nöbeling-Pontryagin theorem''' then states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the [[Euclidean space]]s, with their usual topology. The '''Menger-Nöbeling theorem''' (1932) states that if ''X'' is compact metric separable and of dimension ''n'', then it embeds as a subspace of Euclidean space of dimension 2''n'' + 1. ([[Georg Nöbeling]] was a student of [[Karl Menger]]. He introduced '''Nöbeling space''', the subspace of '''R'''<sup>2''n'' + 1</sup> consisting of points with at least ''n'' + 1 co-ordinates being [[irrational number]]s, which has universal properties for embedding spaces of dimension ''n''.)

			Assuming only ''X'' metrizable we have ([[Miroslav Katětov]])

			:ind ''X'' ≤ Ind ''X'' = dim ''X'';

			or assuming ''X'' [[compact space\|compact]] and [[Hausdorff space\|Hausdorff]] ([[P. S. Aleksandrov]])

			:dim ''X'' ≤ ind ''X'' ≤ Ind ''X''.

			Either inequality here may be strict; an example of Vladimir V. Filippov shows that the two inductive dimensions may differ.

			A separable metric space ''X'' satisfies the inequality <math>\operatorname{Ind}X\le n</math> if and only if for every closed sub-space <math>A</math> of the space <math>X</math> and each continuous mapping <math>f:A\to S^n</math> there exists a continuous extension <math>\bar f:X\to S^n</math>.

			==References==
			{{Empty section\|date=July 2010}}
			==Further reading==
			*Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in [[Ivor Grattan-Guinness\|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 844-55.
			*R. Engelking, ''Theory of Dimensions. Finite and Infinite'', Heldermann Verlag (1995), ISBN 3-88538-010-2.
			*V. V. Fedorchuk, ''The Fundamentals of Dimension Theory'', appearing in ''Encyclopaedia of Mathematical Sciences, Volume 17, General Topology I'', (1993) A. V. Arkhangel'skii and L. S. Pontryagin (Eds.), Springer-Verlag, Berlin ISBN 3-540-18178-4.
			*V. V. Filippov, ''On the inductive dimension of the product of bicompacta'', Soviet. Math. Dokl., 13 (1972), N° 1, 250-254.
			*A. R. Pears, ''Dimension theory of general spaces'', Cambridge University Press (1975).

			[[Category:Dimension theory]]

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2012-07-18T14:35:54Z

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