Regular space

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

Definition

A topological space is said to be regular 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	given any point and a closed subset not containing it, there are disjoint open subsets containing them.	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 $x\in U,A\subseteq V$, and $U\cap V=\varnothing$.
2	open neighborhood contains closure of smaller open neighborhood	given any point and an open subset containing it, there is an open subset containing the point whose closure lies in the original open subset.	given any point $x\in X$ and open subset $V\subseteq X$ such that $x\in V$, there exists an open subset $U\subseteq X$ such that $x\in U$ and the closure ${\overline {U}}$ is contained in $V$.
3	separation of compact subset and closed subset	given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them.	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 $U\cap V=\varnothing$.
4	basis element contains closure of smaller basis element	(fix a choice of basis of open subsets) given any point and a basis open subset containing it, there is a basis open subset containing the point whose closure lies in the original open subset.	given any point $x\in X$ and a basis open subset $V\subseteq X$ such that $x\in V$, there exists a basis open subset $U\subseteq X$ such that $x\in U$ and the closure ${\overline {U}}$ is contained in $V$.
5	basis element contains closure of smaller open subset	(fix a choice of basis of open subsets) given any point and a basis open subset containing it, there is an open subset containing the point whose closure lies in the original open subset.	given any point $x\in X$ and a basis open subset $V\subseteq X$ such that $x\in V$, there exists an open subset $U\subseteq X$ such that $x\in U$ and the closure ${\overline {U}}$ is contained in $V$.
6	open subset contains basis element	(fix a choice of basis of open subsets) given any point and an open subset containing it, there is a basis open subset containing the point whose closure lies in the original open subset.	given any point $x\in X$ and an open subset $V\subseteq X$ such that $x\in V$, there exists a basis open subset $U\subseteq X$ such that $x\in U$ and the closure ${\overline {U}}$ is contained in $V$.

Outside of point-set topology, the term regular space is often used for a regular Hausdorff space, which is the same thing as a regular T1 space. This convention is, however, eschewed by point-set topologists.

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

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
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
semiregular space		regular implies semiregular
locally regular space
preregular space		regular implies preregular
symmetric 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)