Regular space

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

• Metrizable space
• CW-space
• Perfectly normal space
• Hereditarily normal space
• Monotonically normal space
• Normal space
• Completely regular space

Weaker properties

• Hausdorff space
• T1 space
• Kolmogorov space

Metaproperties

Metaproperty name	Satisfied?	Proof	Section in this article
subspace-hereditary property of topological spaces	Yes	regularity is hereditary	#Hereditariness
product-closed property of topological spaces	Yes	regularity is product-closed	#Products
box product-closed property of topological spaces	Yes	regularity is box product-closed	#Box products
refining-preserved property of topological spaces	No	regularity is not refining-preserved	#Refining

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a regular space is regular. For full proof, refer: Regularity is hereditary

Further information: Hausdorffness is hereditary, complete regularity is hereditary, normality is not hereditary

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

An arbitrary product of regular spaces is regular. For full proof, refer: Regularity is product-closed

Box products

This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces

An arbitrary box product of regular spaces is regular. For full proof, refer: Regularity is box-product-closed

Refining

NO: This property of topological spaces is not a refining-preserved property of topological spaces. In other words, putting a finer topology on a topological space satisfying this property might give a topological space not satisfying this property.
View other properties that are not refining-preserved

Moving to a finer topology does not preserve regularity. In other words, if $(X,\tau )$ is a regular space, and ${\tau }^{\prime }$ is a finer topology on $X$ than $\tau$, then $(X,{\tau }^{\prime })$ need not be a regular space. For full proof, refer: Regularity is not refining-preserved

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)