Paracompact Hausdorff space: Difference between revisions

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A [[topological space]] is termed '''paracompact Hausdorff''' if it satisfies the following equivalent conditions:
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.
The second definition is the one used in algebraic topology.
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===Stronger properties===
===Stronger properties===


* [[Weaker than::Compact Hausdorff space]]
{| class="sortable" border="1"
* [[Weaker than::Locally compact paracompact Hausdorff space]]
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Weaker than::Polyhedron]]
|-
* [[Weaker than::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}}
* [[Weaker than::Metrizable space]]
|-
* [[Weaker than::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===
===Weaker properties===


* [[Stronger than::Binormal space]]
{| class="sortable" border="1"
* [[Stronger than::Normal space]]: {{proofat|[[Paracompact Hausdorff implies normal]]}}
! 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}}
|}

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:

  1. It is paracompact and Hausdorff
  2. 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
  3. It is regular and every open cover has a locally finite open refinement
  4. It is regular and every open cover has a locally finite closed refinement
  5. It is regular and every open cover has a locally finite refinement
  6. 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 (via normal Hausdorff) (via normal Hausdorff) Normal Hausdorff space|FULL LIST, MORE INFO
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