Line with two origins - Revision history

Vipul: /* Properties it does not satisfy */

2016-05-31T14:49:56Z

Properties it does not satisfy

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	\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|
	\|-		\|-
			\| [[dissatisfies property::compact space]] \|\| No \|\| \|\|
			\|-
	! colspan="4" style="text-align:center; background: white"\| Countability		! colspan="4" style="text-align:center; background: white"\| Countability
	\|-		\|-

Vipul at 14:49, 31 May 2016

2016-05-31T14:49:24Z

← Older revision		Revision as of 14:49, 31 May 2016
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	! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied		! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
	\|-		\|-
	! Separation		! colspan="4" style="text-align:center; background: white"\| Separation
	\|-		\|-
	\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]		\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]
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	\| [[satisfies property::locally regular space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally regular space]] \|\| Yes \|\| \|\|
	\|-		\|-
	! Connectedness		! colspan="4" style="text-align:center; background: white"\| Connectedness
	\|-		\|-
	\| [[satisfies property::path-connected space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::connected space]]		\| [[satisfies property::path-connected space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::connected space]]
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	\| [[satisfies property::locally contractible space]] \|\| Yes \|\| \|\|satisfies: [[satisfies property::locally path-connected space]], [[satisfies property::locally simply connected space]], [[satisfies property::semilocally simply connected space]], [[satisfies property::locally connected space]]		\| [[satisfies property::locally contractible space]] \|\| Yes \|\| \|\|satisfies: [[satisfies property::locally path-connected space]], [[satisfies property::locally simply connected space]], [[satisfies property::semilocally simply connected space]], [[satisfies property::locally connected space]]
	\|-		\|-
	! Compactness		! colspan="4" style="text-align:center; background: white"\| Compactness
	\|-		\|-
	\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|
	\|-		\|-
	! Countability		! colspan="4" style="text-align:center; background: white"\| Countability
	\|-		\|-
	\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]		\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]
	\|-		\|-
	! Miscellaneous		! colspan="4" style="text-align:center; background: white"\| Miscellaneous
	\|-		\|-
	\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]], [[satisfies property::locally Hausdorff space]], [[satisfies property::locally normal space]], [[satisfies property::locally regular space]]		\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]], [[satisfies property::locally Hausdorff space]], [[satisfies property::locally normal space]], [[satisfies property::locally regular space]]

Vipul: /* Definition */

2016-05-31T06:51:13Z

Definition

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	* It is the [[quotient space]] of the disjoint union of two copies of <math>\R</math>, via the identification of <math>x</math> in the first copy with <math>x</math> in the second copy, for <math>x \ne 0</math>.		* It is the [[quotient space]] of the disjoint union of two copies of <math>\R</math>, via the identification of <math>x</math> in the first copy with <math>x</math> in the second copy, for <math>x \ne 0</math>.
	* It is the real line with ''two'' origins, i.e. with two copies of the origin, wherein although both copies of the origin are separated, arbitrarily small nonzero real numbers approach both these copies.		* It is the real line with ''two'' origins, i.e. with two copies of the origin, wherein although both copies of the origin are separated, arbitrarily small nonzero real numbers approach both these copies.
	* It is the [[topological space]] with an [[étale map]] to the real line, such that the fiber at the origin is a two-point set, and the fiber at any other point of the real line, is a one-point set.		* It is the [[topological space]] with an [[étale map]] to the real line, such that the fiber at the origin is a two-point set, and the fiber at any other point of the real line is a one-point set.

	==Topological space properties==		==Topological space properties==

Vipul: /* Properties it does not satisfy */

2012-01-28T01:55:19Z

Properties it does not satisfy

← Older revision		Revision as of 01:55, 28 January 2012
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	\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]		\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]
	\|-		\|-
	\| [[dissatisfies property::US-space]] \|\| No \|\| \|\| dissatisfies: [[dissatisfies property::KC-space]], [[dissatisfies property::Hausdorff space]], [[dissatisfies property::regular space]] (note, it's T1 so regular would imply Hausdorff), [[dissatisfies property::normal space]], [[dissatisfies property::metrizable space]], [[dissatisfies property::		\| [[dissatisfies property::US-space]] \|\| No \|\| \|\| dissatisfies: [[dissatisfies property::KC-space]], [[dissatisfies property::Hausdorff space]], [[dissatisfies property::regular space]] (note, it's T1 so regular would imply Hausdorff), [[dissatisfies property::normal space]], [[dissatisfies property::metrizable space]], [[dissatisfies property::CW-space]]
	\|-		\|-
	\| [[satisfies property::locally normal space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally normal space]] \|\| Yes \|\| \|\|

Vipul: /* Properties it does not satisfy */

2012-01-28T01:54:57Z

Properties it does not satisfy

← Older revision		Revision as of 01:54, 28 January 2012
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	\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]		\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]
	\|-		\|-
	\| [[dissatisfies property::US-space]] \|\| No \|\| \|\| dissatisfies: [[dissatisfies property::KC-space]], [[dissatisfies property::Hausdorff space]], [[dissatisfies property::regular space]] (note, it's T1 so regular would imply Hausdorff), [[dissatisfies property::normal space]]		\| [[dissatisfies property::US-space]] \|\| No \|\| \|\| dissatisfies: [[dissatisfies property::KC-space]], [[dissatisfies property::Hausdorff space]], [[dissatisfies property::regular space]] (note, it's T1 so regular would imply Hausdorff), [[dissatisfies property::normal space]], [[dissatisfies property::metrizable space]], [[dissatisfies property::
	\|-		\|-
	\| [[satisfies property::locally normal space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally normal space]] \|\| Yes \|\| \|\|

Vipul: /* Properties it does not satisfy */

2012-01-28T01:50:43Z

Properties it does not satisfy

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	! Miscellaneous		! Miscellaneous
	\|-		\|-
	\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]], [[satisfies property::locally Hausdorff space]]		\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]], [[satisfies property::locally Hausdorff space]], [[satisfies property::locally normal space]], [[satisfies property::locally regular space]]
	\|}		\|}

Vipul: /* Properties it does not satisfy */

2012-01-28T01:45:31Z

Properties it does not satisfy

← Older revision		Revision as of 01:45, 28 January 2012
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	! Miscellaneous		! Miscellaneous
	\|-		\|-
	\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]]		\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]], [[satisfies property::locally Hausdorff space]]
	\|}		\|}

Vipul: /* Properties it does not satisfy */

2012-01-28T01:42:18Z

Properties it does not satisfy

← Older revision		Revision as of 01:42, 28 January 2012
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	\|-		\|-
	! Countability		! Countability
			\|-
	\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]		\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]
	\|-		\|-
	! Miscellaneous		! Miscellaneous
			\|-
	\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]]		\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]]
	\|}		\|}

Vipul: /* Properties it does not satisfy */

2012-01-28T01:41:56Z

Properties it does not satisfy

← Older revision		Revision as of 01:41, 28 January 2012
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	\|-		\|-
	\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|		\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|
	\|-		\|-
	\|
	! Countability		! Countability
	\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]		\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]

Vipul: /* Topological space properties */

2012-01-28T01:41:26Z

Topological space properties

← Older revision		Revision as of 01:41, 28 January 2012
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	===Properties it does not satisfy===		===Properties it does not satisfy===

	* [[Hausdorff space]]: ~~The line with two origins is not a Hausdorff~~ space, ~~because we cannot find open sets separating the two origins~~		{\| class="sortable" border="1"
	* [[US-space]]: it is '~~'not'' true for this~~ space ~~that every sequence has at most one limit~~		! Property !! Satisfied? !! Explanation !! Corollary properties satisfied/dissatisfied
			\|-
	~~===Properties it does satisfy===~~		! Separation
			\|-
	* [[~~Locally Euclidean~~ space]]: ~~Every point has a neighbourhood homeomorphic to <math>\R^1</math>~~		\| [[satisfies property::locally Hausdorff space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::T1 space]], [[satisfies property::Kolmogorov space]]
	* [[~~Second~~-countable space]]: ~~There is a~~ countable ~~basis of open sets~~		\|-
			\| [[dissatisfies property::US-space]] \|\| No \|\| \|\| dissatisfies: [[dissatisfies property::KC-space]], [[dissatisfies property::Hausdorff space]], [[dissatisfies property::regular space]] (note, it's T1 so regular would imply Hausdorff), [[dissatisfies property::normal space]]
	~~Thus the line with two origins fails to be a manifold only with respect to the Hausdorff condition. It behaves very differently from what we intuitively expect of manifolds~~		\|-
			\| [[satisfies property::locally normal space]] \|\| Yes \|\| \|\|
			\|-
			\| [[satisfies property::locally regular space]] \|\| Yes \|\| \|\|
			\|-
			! Connectedness
			\|-
			\| [[satisfies property::path-connected space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::connected space]]
			\|-
			\| [[satisfies property::locally contractible space]] \|\| Yes \|\| \|\|satisfies: [[satisfies property::locally path-connected space]], [[satisfies property::locally simply connected space]], [[satisfies property::semilocally simply connected space]], [[satisfies property::locally connected space]]
			\|-
			! Compactness
			\|-
			\| [[satisfies property::locally compact space]] \|\| Yes \|\| \|\|
			\|-
			\|
			! Countability
			\| [[satisfies property::second-countable space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::first-countable space]], [[satisfies property::separable space]]
			\|-
			! Miscellaneous
			\| [[satisfies property::locally Euclidean space]] \|\| Yes \|\| \|\| satisfies: [[satisfies property::locally contractible space]]
			\|}

	==References==		==References==
	===Textbook references===		===Textbook references===
	* {{booklink-defined\|Munkres}}, Page 227, Exercise 5, Chapter 4 (full definition given in exercise, by specifying a basis)		* {{booklink-defined\|Munkres}}, Page 227, Exercise 5, Chapter 4 (full definition given in exercise, by specifying a basis)