Linearly orderable space - Revision history

Vipul at 03:25, 27 January 2012

2012-01-27T03:25:06Z

← Older revision		Revision as of 03:25, 27 January 2012
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	==Relation with other properties==		==Relation with other properties==

	~~===Stronger properties===~~

	* [[Well-orderable space]]

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

	* [[~~Monotonically~~ normal space]]		{\| class="sortable" border="1"
	* [[~~Hereditarily~~ normal space]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Normal~~ space]]		\|-
	* [[~~Regular~~ space]]		\| [[Stronger than::monotonically normal space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|monotonically normal space\|linearly orderable space}}
	* [[Hausdorff space]]		\|-
			\| [[Stronger than::hereditarily normal space]] \|\| every subspace is a [[normal space]] \|\| (via monotonically normal) \|\| (via monotonically normal) \|\| {{intermediate notions short\|hereditarily normal space\|linearly orderable space}}
			\|-
			\| [[Stronger than::normal space]] \|\| <math>T_1</math> and disjoint closed subsets can be separated by disjoint open subsets \|\| (via monotonically normal) \|\| (via monotonically normal) \|\|{{intermediate notions short\|normal space\|linearly orderable space}}
			\|-
			\|[[Stronger than::completely regular space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|completely regular space\|linearly orderable space}}
			\|-
			\| [[Stronger than::regular space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|regular space\|linearly orderable space}}
			\|-
			\| [[Stronger than::Hausdorff space]] \|\| distinct points can be separated by disjoint open subsets \|\| [[linearly orderable implies Hausdorff]], also via others \|\| (via regular, normal) \|\| {{intermediate notions short\|Hausdorff space\|linearly orderable space}}
			\|-
			\| [[Stronger than::T1 space]] \|\| points are closed \|\| [[linearly orderable implies T1]], also via Hausdorff \|\| (via Hausdorff, others) \|\| {{intermediate notions short\|T1 space\|linearly orderable space}}
			\|}

Vipul: 2 revisions

2008-05-11T19:48:39Z

2 revisions

← Older revision	Revision as of 19:48, 11 May 2008
(No difference)

Vipul: /* Weaker properties */

2007-12-17T20:14:53Z

Weaker properties

← Older revision		Revision as of 20:14, 17 December 2007
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	===Weaker properties===		===Weaker properties===

			* [[Monotonically normal space]]
			* [[Hereditarily normal space]]
	* [[Normal space]]		* [[Normal space]]
	* [[Regular space]]		* [[Regular space]]
	* [[Hausdorff space]]		* [[Hausdorff space]]

Vipul at 21:03, 10 November 2007

2007-11-10T21:03:06Z

New page

{{topospace property}}

==Definition==

A [[topological space]] is termed '''linearly orderable''' if it occurs as the underlying topological space of a [[linearly ordered space]] (viz it can be obtained by giving the [[order topology]] to a linearly ordered set).

==Relation with other properties==

===Stronger properties===

* [[Well-orderable space]]

===Weaker properties===

* [[Normal space]]
* [[Regular space]]
* [[Hausdorff space]]