Moore space: Difference between revisions

Revision as of 02:43, 27 January 2012

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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 a Moore space if it is regular and developable.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
metrizable space		metrizable implies Moore		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
developable space
regular space
Hausdorff space

Facts

Traylor's theorem: Every metacompact separable normal Moore space is metrizable.
Reed-Zenor theorem: Every locally compact locally connected normal Moore space is metrizable.
Jones' theorem: If $2^{ℵ_{0}} < 2^{ℵ_{1}}$ then every separable normal Moore space is metrizable (i.e., we can drop the metacompactness assumption from Traylor's theorem).

@@ Line 5: / Line 5: @@
 ==Definition==
-===Symbol-free definition===
+A [[topological space]] is termed a '''Moore space''' if it is [[regular space|regular]] and [[developable space|developable]].
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::metrizable space]] || || [[metrizable implies Moore]] || || {{intermediate notions short|Moore space|metrizable space}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::developable space]] || || || ||
+|-
+| [[Stronger than::regular space]] || || || ||
+|-
+| [[Stronger than::Hausdorff space]] || || || ||
+|}
+==Facts==
-A [[topological space]] is termed a '''Moore space''' if it is [[regular space|regular]] and [[developable space|developable]].
+* [[Traylor's theorem]]: Every [[metacompact space|metacompact]] [[separable space|separable]] [[normal space|normal]] Moore space is metrizable.
+* [[Reed-Zenor theorem]]: Every [[locally compact space|locally compact]] [[locally connected space|locally connected]] [[normal space|normal]] Moore space is metrizable.
+* [[Jones' theorem]]: If <math>2^{\aleph_0} < 2^{\aleph_1}</math> then every [[separable space|separable]] [[normal space|normal]] Moore space is metrizable (i.e., we can drop the metacompactness assumption from Traylor's theorem).