Locally Euclidean space - Revision history

Vipul: /* Manifold properties not satisfied for locally Euclidean spaces */

2016-05-30T21:25:34Z

Manifold properties not satisfied for locally Euclidean spaces

← Older revision		Revision as of 21:25, 30 May 2016
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	\| [[line with two origins]] \|\| [[Hausdorff space]] \|\| No \|\| Yes \|\| 1 \|\|		\| [[line with two origins]] \|\| [[Hausdorff space]] \|\| No \|\| Yes \|\| 1 \|\|
	\|-		\|-
	\| [[Prufer manifold \|\| [[normal space]] \|\| Yes \|\| No \|\| 2 \|\|		\| [[Prufer manifold]] \|\| [[normal space]] \|\| Yes \|\| No \|\| 2 \|\|
	\|-		\|-
	\| [[long line]] \|\| [[metrizable space]] \|\| Yes \|\| No \|\| 1 \|\| [[normal space]]		\| [[long line]] \|\| [[metrizable space]] \|\| Yes \|\| No \|\| 1 \|\| [[normal space]]

Vipul: /* Manifold properties not satisfied for locally Euclidean spaces */

2016-05-30T21:25:24Z

Manifold properties not satisfied for locally Euclidean spaces

← Older revision		Revision as of 21:25, 30 May 2016
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	===Manifold properties not satisfied for locally Euclidean spaces===		===Manifold properties not satisfied for locally Euclidean spaces===

	* [[Hausdorff ~~space]]: The~~ [[line with two origins]] ~~is an example of a locally 1-Euclidean space that is second-countable but not Hausdorff~~		{\| class="sortable" border="1"
	* [[~~Normal~~ space]]~~: The~~ [[Prufer manifold]] ~~is an example of a locally~~ 2-~~Euclidean space that is Hausdorff but not normal (it also fails to be second-countable)~~		! Example !! Property failed !! Hausdorff? !! Second-countable? !! Dimension !! Other intermediate properties (that are always satisfied by manifold) that it satisfies
	* [[~~Metrizable space~~]]~~: The~~ [[~~long line~~]] ~~is an example of a Hausdorff, locally~~ 1~~-Euclidean~~ space ~~that is Hausdorff and in fact normal but not metrizable (it also fails to be second~~-~~countable).~~		\|-
	* [[~~Manifold~~]]~~: The~~ [[~~dictionary plane~~]] ~~is an example of a~~ metrizable ~~locally 2-Euclidean~~ space ~~that is not a manifold (it fails to be second-countable)~~		\| [[line with two origins]] \|\| [[Hausdorff space]] \|\| No \|\| Yes \|\| 1 \|\|
			\|-
			\| [[Prufer manifold \|\| [[normal space]] \|\| Yes \|\| No \|\| 2 \|\|
			\|-
			\| [[long line]] \|\| [[metrizable space]] \|\| Yes \|\| No \|\| 1 \|\| [[normal space]]
			\|-
			\| [[dictionary plane]] \|\| [[manifold]] \|\| Yes \|\| No \|\| 1 \|\| [[metrizable space]], [[normal space]]
			\|}

Vipul: /* Manifold properties not satisfied for locally Euclidean spaces */

2016-05-30T21:19:46Z

Manifold properties not satisfied for locally Euclidean spaces

← Older revision		Revision as of 21:19, 30 May 2016
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	* [[Normal space]]: The [[Prufer manifold]] is an example of a locally 2-Euclidean space that is Hausdorff but not normal (it also fails to be second-countable)		* [[Normal space]]: The [[Prufer manifold]] is an example of a locally 2-Euclidean space that is Hausdorff but not normal (it also fails to be second-countable)
	* [[Metrizable space]]: The [[long line]] is an example of a Hausdorff, locally 1-Euclidean space that is Hausdorff and in fact normal but not metrizable (it also fails to be second-countable).		* [[Metrizable space]]: The [[long line]] is an example of a Hausdorff, locally 1-Euclidean space that is Hausdorff and in fact normal but not metrizable (it also fails to be second-countable).
	* [[Manifold]]: The dictionary plane is an example of a metrizable locally 2-Euclidean space that is not a manifold (it fails to be second-countable)		* [[Manifold]]: The [[dictionary plane]] is an example of a metrizable locally 2-Euclidean space that is not a manifold (it fails to be second-countable)

Vipul: /* Relation with other properties */

2016-05-30T21:17:24Z

Relation with other properties

← Older revision		Revision as of 21:17, 30 May 2016
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	===Weaker properties===		===Weaker properties===

	* [[~~Locally~~ contractible space]]		{\| class="sortable" border="1"
	* [[~~Locally~~ path-connected space]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Locally~~ metrizable space]]		\|-
	* [[~~Locally~~ normal space]]		\| [[Stronger than::locally contractible space]] \|\| has a basis of contractible spaces \|\| follows from the fact that Euclidean implies contractible \|\| a pair of intersecting lines is locally contractible but not locally Euclidean (at the point of intersection) \|\| {{intermediate notions short\|locally contractible space\|locally Euclidean space}}
	* [[~~Locally~~ Hausdorff space]]		\|-
			\| [[Stronger than::locally path-connected space]] \|\| has a basis of path-connected spaces \|\| (via locally contractible) \|\| (via locally contractible) \|\| {{intermediate notions short\|locally path-connected space\|locally Euclidean space}}
			\|-
			\| [[Stronger than::locally metrizable space]] \|\| every point is contained in a metrizable open subset \|\| follows from the fact that Euclidean implies metrizable, using the Euclidean metric \|\| the pair of intersecting lines is metrizable (and hence locally metrizable) but not locally Eucldiean \|\| {{intermediate notions short\|locally metrizable space\|locally Euclidean space}}
			\|-
			\| [[Stronger than::locally normal space]] \|\| has a basis of normal spaces \|\| (via locally metrizable) \|\| (via locally metrizable) \|\| {{intermediate notions short\|locally normal space\|locally Euclidean space}}
			\|-
			\| [[Stronger than::locally Hausdorff space]] \|\| every point is contained in a Hausdorff open subset \|\| (via locally metrizable) \|\| (via locally metrizable) \|\| {{intermediate notions short\|locally Hausdorff space\|locally Euclidean space}}
			\|}

	===Manifold properties not satisfied for locally Euclidean spaces===		===Manifold properties not satisfied for locally Euclidean spaces===

Vipul: /* Formalisms */

2016-05-30T21:10:00Z

Formalisms

← Older revision		Revision as of 21:10, 30 May 2016
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	{{obtained by applying the\|locally operator\|Euclidean space}}		{{obtained by applying the\|locally operator\|Euclidean space}}

	The equivalence between multiple sense of the term "locally" follows from the fact that any Euclidean space is [[self-based space\|self-based]]		The equivalence between multiple sense of the term "locally" follows from the fact that any Euclidean space is [[self-based space\|self-based]].

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2016-05-30T21:09:54Z

Definition

← Older revision		Revision as of 21:09, 30 May 2016
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	This case could arise if the space has multiple connected components that have different dimensions. For instance, a disjoint union of a plane and a line is locally 2-Euclidean at the points on the plane and locally Euclidean at the points on the line.		This case could arise if the space has multiple connected components that have different dimensions. For instance, a disjoint union of a plane and a line is locally 2-Euclidean at the points on the plane and locally Euclidean at the points on the line.

			== Formalisms ==

			{{obtained by applying the\|locally operator\|Euclidean space}}

			The equivalence between multiple sense of the term "locally" follows from the fact that any Euclidean space is [[self-based space\|self-based]]

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Stronger properties */

2016-05-30T21:09:10Z

Stronger properties

← Older revision		Revision as of 21:09, 30 May 2016
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	===Stronger properties===		===Stronger properties===

	* [[~~Manifold~~]]~~: For a manifold, we~~ assume additionally the conditions of [[Hausdorff space\|Hausdorff]] and [[second-countable space\|second-countable]]. The [[line with two origins]] is an example of a locally Euclidean space which is not a manifold, and also shows how many properties that we prove for manifolds, fail to hold for arbitrary locally Euclidean spaces.		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::manifold]] \|\| We assume additionally the conditions of [[Hausdorff space\|Hausdorff]] and [[second-countable space\|second-countable]]. We also require the dimension to be the same at all points. \|\| (obvious) \|\| The [[line with two origins]] is an example of a locally Euclidean space which is not a manifold, and also shows how many properties that we prove for manifolds, fail to hold for arbitrary locally Euclidean spaces. \|\| {{intermediate notions short\|locally Euclidean space\|manifold}}
			\|}

	===Weaker properties===		===Weaker properties===

Vipul: /* Definition */

2016-05-30T21:07:37Z

Definition

← Older revision		Revision as of 21:07, 30 May 2016
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	==Definition==		==Definition==

			=== Locally Euclidean of a fixed dimension ===

	A [[topological space]] <math>X</math> is termed '''locally <math>m</math>-Euclidean''' for a nonnegative integer <math>m</math> such that it satisfies the following equivalent conditions:		A [[topological space]] <math>X</math> is termed '''locally <math>m</math>-Euclidean''' for a nonnegative integer <math>m</math> such that it satisfies the following equivalent conditions:
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	The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.		The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.

			=== Locally Euclidean of possibly varying dimension ===

	The term '''locally Euclidean''' is also sometimes used in the case where we allow the <math>m</math> to vary with the point. In other words, the equivalent conditions (1)-(3) must hold, but the nonnegative integer could vary with the point.		The term '''locally Euclidean''' is also sometimes used in the case where we allow the <math>m</math> to vary with the point. In other words, the equivalent conditions (1)-(3) must hold, but the nonnegative integer could vary with the point.

Vipul: /* Definition */

2016-05-30T21:07:02Z

Definition

← Older revision		Revision as of 21:07, 30 May 2016
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	The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.		The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.

	The term '''locally Euclidean''' is also sometimes used in the case where we allow the <math>m</math> to vary with the point. This case could arise if the space has multiple connected components that have different dimensions.		The term '''locally Euclidean''' is also sometimes used in the case where we allow the <math>m</math> to vary with the point. In other words, the equivalent conditions (1)-(3) must hold, but the nonnegative integer could vary with the point.

			This case could arise if the space has multiple connected components that have different dimensions. For instance, a disjoint union of a plane and a line is locally 2-Euclidean at the points on the plane and locally Euclidean at the points on the line.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2016-05-30T21:05:37Z

Definition

← Older revision		Revision as of 21:05, 30 May 2016
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	==Definition==		==Definition==

	A [[topological space]] <math>X</math> is termed '''locally Euclidean''' ~~if there exists~~ a nonnegative integer <math>m</math> such that it satisfies the following equivalent conditions:		A [[topological space]] <math>X</math> is termed '''locally <math>m</math>-Euclidean''' for a nonnegative integer <math>m</math> such that it satisfies the following equivalent conditions:

	# For any point <math>x \in X</math>, there exists an open subset <math>U \subseteq X</math> such that <math>x \in U</math>, and <math>U</math> is homeomorphic to the [[Euclidean space]] <math>\R^m</math>.		# For any point <math>x \in X</math>, there exists an open subset <math>U \subseteq X</math> such that <math>x \in U</math>, and <math>U</math> is homeomorphic to the [[Euclidean space]] <math>\R^m</math>.
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	The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.		The equivalence of the three definitions follows from the fact that any Euclidean space is [[self-based space\|self-based]]: it has a basis of open subsets all of which are homeomorphic to the whole space.

			The term '''locally Euclidean''' is also sometimes used in the case where we allow the <math>m</math> to vary with the point. This case could arise if the space has multiple connected components that have different dimensions.

	==Relation with other properties==		==Relation with other properties==