Linearly orderable space: Difference between revisions

Latest revision as of 03:25, 27 January 2012

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
monotonically normal space				\|FULL LIST, MORE INFO
hereditarily normal space	every subspace is a normal space	(via monotonically normal)	(via monotonically normal)	Hereditarily collectionwise normal space, Monotonically normal space\|FULL LIST, MORE INFO
normal space	$T_{1}$ and disjoint closed subsets can be separated by disjoint open subsets	(via monotonically normal)	(via monotonically normal)	Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space, Monotonically normal space\|FULL LIST, MORE INFO
completely regular space				Monotonically normal space, Normal Hausdorff space\|FULL LIST, MORE INFO
regular space				Monotonically normal space, Normal Hausdorff space\|FULL LIST, MORE INFO
Hausdorff space	distinct points can be separated by disjoint open subsets	linearly orderable implies Hausdorff, also via others	(via regular, normal)	Monotonically normal space, Normal Hausdorff space\|FULL LIST, MORE INFO
T1 space	points are closed	linearly orderable implies T1, also via Hausdorff	(via Hausdorff, others)	Normal Hausdorff space\|FULL LIST, MORE INFO

@@ Line 6: / Line 6: @@
 ==Relation with other properties==
-===Stronger properties===
-* [[Well-orderable space]]
 ===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}}
+|}