CW-space: Difference between revisions

Latest revision as of 20:13, 29 January 2014

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 said to be a CW-space if it possesses a CW-decomposition, or in other words, if it can be viewed as the underlying topological space of a CW-complex.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Polyhedron	occurs as the geometric realization of a simplicial complex			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hereditarily paracompact Hausdorff space	Hausdorff space that is also a hereditarily paracompact space: every subspace (with the subspace topology) is a paracompact space			\|FULL LIST, MORE INFO
Paracompact Hausdorff space		CW implies paracompact Hausdorff		\|FULL LIST, MORE INFO
Perfectly normal space		CW implies perfectly normal		\|FULL LIST, MORE INFO
Normal space		CW implies normal		Hereditarily normal space, Paracompact Hausdorff space, Perfectly normal space\|FULL LIST, MORE INFO
Hausdorff space		CW implies Hausdorff		Completely regular space, Normal Hausdorff space, Regular Hausdorff space\|FULL LIST, MORE INFO
Locally contractible space	every point is contained in a contractible open subset			\|FULL LIST, MORE INFO
Locally path-connected space		[[CW implies locally path-connected]		Locally contractible space\|FULL LIST, MORE INFO
Homotopy-CW-space	homotopy-equivalent to a CW-space	(obvious)	any contractible space that is not a Hausdorff space, e.g., the line with two origins	\|FULL LIST, MORE INFO

@@ Line 9: / Line 9: @@
 ===Stronger properties===
-* [[Polyhedron]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::Polyhedron]] || occurs as the geometric realization of a [[simplicial complex]] || || || {{intermediate notions short|CW-space|polyhedron}}
+|}
 ===Weaker properties===
-* [[Hereditarily paracompact Hausdorff space]]
+{| class="sortable" border="1"
-* [[Paracompact Hausdorff space]]: {{proofat|[[CW implies paracompact Hausdorff]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Perfectly normal space]]: {{proofat|[[CW implies perfectly normal]]}}
+|-
-* [[Normal space]]: {{proofat|[[CW implies normal]]}}
+| [[Stronger than::Hereditarily paracompact Hausdorff space]] || [[Hausdorff space]] that is also a [[hereditarily paracompact space]]: every subspace (with the [[subspace topology]]) is a [[paracompact space]] || || || {{intermediate notions short|hereditarily paracompact Hausdorff space|CW-space}}
-* [[Hausdorff space]]
+|-
-* [[Locally contractible space]]
+| [[Stronger than::Paracompact Hausdorff space]] || || [[CW implies paracompact Hausdorff]] || || {{intermediate notions short|paracompact Hausdorff space|CW-space}}
-* [[Locally path-connected space]]: {{proofat|[[CW implies locally path-connected]]}}
+|-
-* [[Homotopy-CW-space]]
+| [[Stronger than::Perfectly normal space]] || || [[CW implies perfectly normal]] || || {{intermediate notions short|perfectly normal space|CW-space}}
+|-
+| [[Stronger than::Normal space]] || || [[CW implies normal]] || || {{intermediate notions short|normal space|CW-space}}
+|-
+| [[Stronger than::Hausdorff space]] || || [[CW implies Hausdorff]] || || {{intermediate notions short|Hausdorff space|CW-space}}
+|-
+| [[Stronger than::Locally contractible space]] || every point is contained in a contractible open subset || || || {{intermediate notions short|locally contractible space|CW-space}}
+|-
+| [[Stronger than::Locally path-connected space]] || || [[CW implies locally path-connected] || || {{intermediate notions short|locally path-connected space|CW-space}}
+|-
+| [[Stronger than::Homotopy-CW-space]] || [[homotopy-equivalent topological spaces|homotopy-equivalent]] to a CW-space || (obvious) || any contractible space that is not a Hausdorff space, e.g., the [[line with two origins]] || {{intermediate notions short|homotopy-CW-space|CW-space}}
+|}