Regular Hausdorff space

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