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
Relation with other properties
Stronger properties
| Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
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| metrizable space |
underlying topological space of a metric space |
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Tychonoff space|FULL LIST, MORE INFO
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| CW-space |
topological space arising as the underlying space of a CW-complex |
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|FULL LIST, MORE INFO
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| perfectly normal Hausdorff space |
it is normal and every closed subset is a G-delta subset |
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|FULL LIST, MORE INFO
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| hereditarily normal Hausdorff space |
every subset is normal in the subspace topology |
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|FULL LIST, MORE INFO
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| monotonically normal Hausdorff space |
(follow link for definition) |
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|FULL LIST, MORE INFO
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| normal Hausdorff space |
T1 and disjoint closed subsets can be separated by disjoint open subsets |
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Tychonoff space|FULL LIST, MORE INFO
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| Tychonoff space |
T1 and point and closed subset not containing it can be separated by continuous function |
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|FULL LIST, MORE INFO
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| compact Hausdorff space |
compact and Hausdorff |
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Tychonoff space|FULL LIST, MORE INFO
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| locally compact Hausdorff space |
locally compact and Hausdorff |
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|FULL LIST, MORE INFO
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| paracompact Hausdorff space |
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|FULL LIST, MORE INFO
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