Regular Hausdorff space: Difference between revisions
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{{T family|T3}} | {{T family|T3}} | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::metrizable space]] || underlying topological space of a [[metric space]] || || || {{intermediate notions short|regular Hausdorff space|metrizable space}} | |||
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| [[Weaker than::CW-space]] || topological space arising as the underlying space of a [[CW-complex]] || || || {{intermediate notions short|regular Hausdorff space|CW-space}} | |||
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| [[Weaker than::perfectly normal Hausdorff space]] || it is [[normal space|normal]] and every [[closed subset]] is a [[G-delta subset]] || || || {{intermediate notions short|regular Hausdorff space|perfectly normal Hausdorff space}} | |||
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| [[Weaker than::hereditarily normal Hausdorff space]] || every subset is normal in the subspace topology || || || {{intermediate notions short|regular Hausdorff space|hereditarily normal Hausdorff space}} | |||
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| [[Weaker than::monotonically normal Hausdorff space]] || (follow link for definition) || || || {{intermediate notions short|regular Hausdorff space|monotonically normal Hausdorff space}} | |||
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| [[Weaker than::normal Hausdorff space]] || T1 and disjoint closed subsets can be separated by disjoint open subsets || || || {{intermediate notions short|regular Hausdorff space|normal Hausdorff space}} | |||
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| [[Weaker than::Tychonoff space]] || T1 and point and closed subset not containing it can be separated by continuous function || || || {{intermediate notions short|regular Hausdorff space|Tychonoff space}} | |||
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| [[Weaker than::compact Hausdorff space]] || [[compact space|compact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|regular Hausdorff space|compact Hausdorff space}} | |||
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| [[Weaker than::locally compact Hausdorff space]] || [[locally compact space|locally compact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|regular Hausdorff space|locally compact Hausdorff space}} | |||
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| [[Weaker than::paracompact Hausdorff space]] || || || || {{intermediate notions short|regular Hausdorff space|paracompact Hausdorff space}} | |||
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Revision as of 21:49, 27 January 2012
Definition
A topological space is termed a regular Hausdorff space or a space if it satisfies the following equivalent conditions:
- It is both a regular space and a Hausdorff space.
- It is both a regular space and a T1 space.
Note that outside of point-set topology, and in many elementary treatments, the term regular space is used to stand for regular Hausdorff space.
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
In the T family (properties of topological spaces related to separation axioms), this is called: T3