Paracompact Hausdorff space: Difference between revisions

Latest revision as of 23:41, 4 January 2017

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

This article describes a property of topological spaces obtained as a conjunction of the following two properties: paracompactness and Hausdorffness

Definition

A topological space is termed paracompact Hausdorff if it satisfies the following equivalent conditions:

It is paracompact and Hausdorff
Given any open cover of the space, there is a partition of unity subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
It is regular and every open cover has a locally finite open refinement
It is regular and every open cover has a locally finite closed refinement
It is regular and every open cover has a locally finite refinement
It is regular and every open cover has a countably locally finite open refinement

The second definition is the one used in algebraic topology.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
compact Hausdorff space	compact and Hausdorff	compact implies paracompact	paracompact Hausdorff not implies compact	\|FULL LIST, MORE INFO
locally compact paracompact Hausdorff space	paracompact Hausdorff and locally compact			\|FULL LIST, MORE INFO
polyhedron	the underlying topological space of a simplicial complex			CW-space\|FULL LIST, MORE INFO
CW-space	the underlying topological space of a CW-complex	CW implies paracompact Hausdorff		\|FULL LIST, MORE INFO
metrizable space	the underlying topological space of a metric space	metrizable implies paracompact Hausdorff		Elastic space\|FULL LIST, MORE INFO
manifold

Weaker properties

Property	Meaning	Proof of implication	proof of strictness (reverse implication failure)	intermediate notions
binormal space	product with the unit interval is normal	paracompact Hausdorff implies binormal		\|FULL LIST, MORE INFO
normal space	any two disjoint closed subsets are separated by disjoint open subsets	paracompact Hausdorff implies normal	normal not implies paracompact	Binormal space\|FULL LIST, MORE INFO
normal Hausdorff space	normal and a Hausdorff space	paracompact Hausdorff implies normal	normal not implies paracompact	\|FULL LIST, MORE INFO
Collectionwise normal space	any discrete collection of closed subsets can be separated by pairwise disjoint open subsets	paracompact Hausdorff implies collectionwise normal	collectionwise normal not implies paracompact Hausdorff	\|FULL LIST, MORE INFO
Tychonoff space	it is T1 and completely regular	(via normal Hausdorff)	(via normal Hausdorff)	\|FULL LIST, MORE INFO
completely regular space	any point and disjoint closed subset can be separated by a continuous function to $[0,1]$	(via normal Hausdorff)	(via normal Hausdorff)	Normal Hausdorff space\|FULL LIST, MORE INFO

@@ Line 7: / Line 7: @@
 A [[topological space]] is termed '''paracompact Hausdorff''' if it satisfies the following equivalent conditions:
-* It is [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]]
+# It is [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]]
-* Given any open cover of the space, there is a [[partition of unity]] subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
+# Given any open cover of the space, there is a [[partition of unity]] subordinate to that open cover; in other words, there is a partition of unity such that the support of each function is contained in some set of that open cover
-* It is [[regular space|regular]] and every open cover has a [[locally finite collection of subsets|locally finite]] open [[refinement]]
+# It is [[regular space|regular]] and every open cover has a [[locally finite collection of subsets|locally finite]] open [[refinement]]
-* It is [[regular space|regular]] and every open cover has a locally finite closed refinement
+# It is [[regular space|regular]] and every open cover has a locally finite closed refinement
-* It is [[regular space|regular]] and every open cover has a locally finite refinement
+# It is [[regular space|regular]] and every open cover has a locally finite refinement
-* It is [[regular space|regular]] and every open cover has a [[countable locally finite collection of subsets|countably locally finite]] open refinement
+# It is [[regular space|regular]] and every open cover has a [[countable locally finite collection of subsets|countably locally finite]] open refinement
 The second definition is the one used in algebraic topology.
@@ Line 20: / Line 20: @@
 ===Stronger properties===
-* [[Compact Hausdorff space]]
+{| class="sortable" border="1"
-* [[Locally compact paracompact Hausdorff space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Polyhedron]]
+|-
-* [[CW-space]]: {{proofat|[[CW implies paracompact Hausdorff]]}}
+| [[Weaker than::compact Hausdorff space]] || [[compact space|compact]] and Hausdorff || [[compact implies paracompact]] || [[paracompact Hausdorff not implies compact]] || {{intermediate notions short|paracompact Hausdorff space|compact Hausdorff space}}
-* [[Manifold]]
+|-
+| [[Weaker than::locally compact paracompact Hausdorff space]] || paracompact Hausdorff and [[locally compact space|locally compact]] || || || {{intermediate notions short|paracompact Hausdorff space|locally compact paracompact Hausdorff space}}
+|-
+| [[Weaker than::polyhedron]] || the underlying topological space of a [[simplicial complex]] || || || {{intermediate notions short|paracompact Hausdorff space|polyhedron}}
+|-
+| [[Weaker than::CW-space]] || the underlying topological space of a [[CW-complex]] || [[CW implies paracompact Hausdorff]] || || {{intermediate notions short|paracompact Hausdorff space|CW-space}}
+|-
+| [[Weaker than::metrizable space]] || the underlying topological space of a [[metric space]] || [[metrizable implies paracompact Hausdorff]] || || {{intermediate notions short|paracompact Hausdorff space|metrizable space}}
+|-
+| [[Weaker than::manifold]] ||  || || ||
+|}
 ===Weaker properties===
-* [[Binormal space]]
+{| class="sortable" border="1"
-* [[Normal space]]
+! Property !! Meaning !! Proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+|-
+| [[Stronger than::binormal space]] || [[product topology|product]] with the unit interval is [[normal space|normal]] || [[paracompact Hausdorff implies binormal]] || || {{intermediate notions short|binormal space|paracompact Hausdorff space}}
+|-
+| [[Stronger than::normal space]] || any two disjoint closed subsets are separated by disjoint open subsets || [[paracompact Hausdorff implies normal]] || [[normal not implies paracompact]] || {{intermediate notions short|normal space|paracompact Hausdorff space}}
+|-
+| [[Stronger than::normal Hausdorff space]] || normal and a Hausdorff space || [[paracompact Hausdorff implies normal]] || [[normal not implies paracompact]] || {{intermediate notions short|normal Hausdorff space|paracompact Hausdorff space}}
+|-
+| [[Stronger than::Collectionwise normal space]] || any discrete collection of closed subsets can be separated by pairwise disjoint open subsets || [[paracompact Hausdorff implies collectionwise normal]] || [[collectionwise normal not implies paracompact Hausdorff]] || {{intermediate notions short|collectionwise normal space|paracompact Hausdorff space}}
+|-
+| [[Stronger than::Tychonoff space]] || it is [[T1 space|T1]] and [[completely regular space|completely regular]] || (via normal Hausdorff) || (via normal Hausdorff) || {{intermediate notions short|Tychonoff space|paracompact Hausdorff space}}
+|-
+| [[Stronger than::completely regular space]] || any point and disjoint closed subset can be separated by a continuous function to <math>[0,1]</math> || (via normal Hausdorff) || (via normal Hausdorff) || {{intermediate notions short|completely regular space|paracompact Hausdorff space}}
+|}