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

Weaker properties

• Hausdorff space
• T1 space
• Kolmogorov space

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)