Regular Hausdorff space

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Definition

A topological space is termed a regular Hausdorff space or a T_3 space if it satisfies the following equivalent conditions:

  1. It is both a regular space and a Hausdorff space.
  2. It is both a regular space and a T1 space.
  3. It is both a regular space and a Kolmogorov space (i.e., a T_0 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
metrizable space underlying topological space of a metric space Tychonoff space|FULL LIST, MORE INFO
CW-space topological space arising as the underlying space of a CW-complex |FULL LIST, MORE INFO
perfectly normal Hausdorff space it is normal and every closed subset is a G-delta subset |FULL LIST, MORE INFO
hereditarily normal Hausdorff space every subset is normal in the subspace topology |FULL LIST, MORE INFO
monotonically normal Hausdorff space (follow link for definition) |FULL LIST, MORE INFO
normal Hausdorff space T1 and disjoint closed subsets can be separated by disjoint open subsets Tychonoff space|FULL LIST, MORE INFO
Tychonoff space T1 and point and closed subset not containing it can be separated by continuous function |FULL LIST, MORE INFO
compact Hausdorff space compact and Hausdorff Tychonoff space|FULL LIST, MORE INFO
locally compact Hausdorff space locally compact and Hausdorff |FULL LIST, MORE INFO
paracompact Hausdorff space |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
regular space |FULL LIST, MORE INFO
Hausdorff space Urysohn space|FULL LIST, MORE INFO
T1 space Hausdorff space, Urysohn space|FULL LIST, MORE INFO
Kolmogorov space Hausdorff space, Urysohn space|FULL LIST, MORE INFO