Regular space

There are two alternative definitions of the term. Please see: Convention:Hausdorffness assumption

Definition

A topological space is said to be regular or $T_3$ if it satisfies the following equivalent conditions:

No.	Shorthand	A topological space is said to be regular if ...	A topological space $X$ is said to be regular if ...
1	separation of point and closed subset not containing it	all points in it are closed sets, and given any point and a closed subset not containing it, there are disjoint open subsets containing them.	for all $x\in X$, the set $\{x\}$ is closed in $X$, and given any point $x\in X$ and closed subset $A\subseteq X$ such that $x\notin A$, there exist disjoint open subsets $U,V$ of $X$ such that $A\subseteq U,x\in V$, and $U\cap V=\varnothing$.
2	separation of compact subset and closed subset	all points in it are closed sets, and given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them.	for all $x\in X$, the set $\{x\}$ is closed in $X$, and given any two subsets $A,B\subseteq X$ such that $A\cap B=\varnothing$, $A$ is compact and $B$ is closed, there exist disjoint open subsets $U,V$ of $X$ such that $A\subseteq U,B\subseteq V$, and $U\cap V=\varnothing$.

The term regular is sometimes used without the T1 space assumption. This gives a different, weaker notion of regularity, which is referred to here as regular-minus-Hausdorff space. This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T3

This article is about a basic definition in topology.
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Relation with other properties

Conjunction with other properties

• Regular Lindelof space: Conjunction with the property of being a Lindelof space.

Stronger properties

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Hausdorff space	any two distinct points can be separated by disjoint open subsets			|FULL LIST, MORE INFO
T1 space	points are closed			|FULL LIST, MORE INFO
Kolmogorov space				|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subspace-hereditary property of topological spaces	Yes	regularity is hereditary	If $X$ is a regular space and $A$ is a subset of $X$, then $A$ is also a regular space under the subspace topology.
product-closed property of topological spaces	Yes	regularity is product-closed	If $X_i,i\in I$ is a collection of regular spaces, then the product space ${\prod }_{i\in I}{X}_i$ is also regular with the product topology.
box product-closed property of topological spaces	Yes	regularity is box product-closed	If $X_i,i\in I$ is a collection of regular spaces, then the product space ${\prod }_{i\in I}{X}_i$ is also regular with the box topology.
refining-preserved property of topological spaces	No	regularity is not refining-preserved	It is possible to have a topological space that is regular but such that passing to a finer topology gives a topological space that is not regular.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 195, Chapter 4, Section 31 (formal definition, along with normal space)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 28 (formal definition)