Metacompact space - Revision history

Vipul: /* Relation with other properties */

2009-10-25T15:45:56Z

Relation with other properties

← Older revision		Revision as of 15:45, 25 October 2009
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	===Stronger properties===		===Stronger properties===

	* [[Compact space]]		{\| class="wikitable" border="1"
	* [[Paracompact space]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
			\|-
			\| [[Weaker than::Compact space]] \|\| every open cover has a finite subcover \|\| \|\| \|\| {{intermediate notions short\|metacompact space\|compact space}}
			\|-
			\| [[Weaker than::Paracompact space]] \|\| every open cover has a locally finite open refinement \|\| [[paracompact implies metacompact]] \|\| [[metacompact not implies paracompact]] \|\| {{intermediate notions short\|metacompact space\|paracompact space}}
			\|-
			\| [[Weaker than::Hereditarily metacompact space]] \|\| every subspace is metacompact \|\| \|\| [[metacompactness is not hereditary]] \|\| {{intermediate notions short\|metacompact space\|hereditarily metacompact space}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[Orthocompact space]]		{\| class="wikitable" border="1"
	* [[Countably metacompact space]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	* [[MetaLindelof space]]		\|-
	* [[Nearly metacompact space]]		\| [[Stronger than::Orthocompact space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|orthocompact space\|metacompact space}}
			\|-
			\| [[Stronger than::Countably metacompact space]] \|\| every countable open cover has a point-finite open refinement \|\| \|\| [[countably metacompact not implies metacompact]] \|\| {{intermediate notions short\|countably metacompact space\|metacompact space}}
			\|-
			\| [[Stronger than::MetaLindelof space]] \|\| every open cover has a point-countable open refinement \|\| \|\| \|\|
			\|-
			\| [[Stronger than::Nearly metacompact space]] \|\| \|\| \|\| \|\|
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: 3 revisions

2008-05-11T19:53:28Z

3 revisions

← Older revision	Revision as of 19:53, 11 May 2008
(No difference)

Vipul at 22:05, 21 April 2008

2008-04-21T22:05:03Z

← Older revision		Revision as of 22:05, 21 April 2008
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	A direct product of metacompact spaces need not be metacompact. However, it follows from the [[tube lemma]] that a direct product of a metacompact space with a compact space is metacompact.		A direct product of metacompact spaces need not be metacompact. However, it follows from the [[tube lemma]] that a direct product of a metacompact space with a compact space is metacompact.

			==References==
			===Textbook references===
			* {{booklink\|Willard}}, Page 152 (formal definition)

Vipul: /* Weaker properties */

2007-08-19T03:34:58Z

Weaker properties

← Older revision		Revision as of 03:34, 19 August 2007
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	* [[Orthocompact space]]		* [[Orthocompact space]]
			* [[Countably metacompact space]]
			* [[MetaLindelof space]]
			* [[Nearly metacompact space]]

	==Metaproperties==		==Metaproperties==

Vipul at 06:21, 18 August 2007

2007-08-18T06:21:12Z

New page

{{topospace property}}

{{variationof|compactness}}

==Definition==

A [[topological space]] is said to be '''metacompact''' if it satisfies the following property: every open [[cover]] has a [[point-finite collection|point-finite]] open [[refinement]].

==Relation with other properties==

===Stronger properties===

* [[Compact space]]
* [[Paracompact space]]

===Weaker properties===

* [[Orthocompact space]]

==Metaproperties==

{{not DP-closed}}

A direct product of metacompact spaces need not be metacompact. However, it follows from the [[tube lemma]] that a direct product of a metacompact space with a compact space is metacompact.