regular space

Definition 1.

A topological space is a regular space if it is both a $T_{0}$ space (http://planetmath.org/T0Space) and a $T_{3}$ space (http://planetmath.org/T3Space).

Example. Consider the set $\mathbbmss{R}$ with the topology $\sigma$ generated by the basis

$\beta=\{U=V-C:V\mbox{ is open with the standard topology and\ }C\mbox{ is (% infinite) numerable}\}.$

Since $\mathbbmss{Q}$ is numerable and $\mathbbmss{R}$ open, the set of irrational numbers $\mathbbmss{R}-\mathbbmss{Q}$ is open and therefore $\mathbbmss{Q}$ is closed. It can be shown that $\mathbbmss{R}-\mathbbmss{Q}$ is an open set with this topology and $\mathbbmss{Q}$ is closed.

Take any irrational number $x$ . Any open set $V$ containing all $\mathbbmss{Q}$ must contain also $x$ , so the regular space property cannot be satisfied. Therefore, $(\mathbbmss{R},\sigma)$ is not a regular space.

Note

In topology, the terminology for separation axioms is not standard. Therefore there are also other meanings of regular. In some references (e.g. [2]) the meanings of regular and $T_{3}$ is exchanged. That is, $T_{3}$ is a stronger property than regular.

References

1 L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.
2 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.

Title	regular space
Canonical name	RegularSpace
Date of creation	2013-03-22 12:18:21
Last modified on	2013-03-22 12:18:21
Owner	drini (3)
Last modified by	drini (3)
Numerical id	11
Author	drini (3)
Entry type	Definition
Classification	msc 54D10
Synonym	regular
Related topic	SeparationAxioms
Related topic	T0Space
Related topic	T2Space
Related topic	T3Space
Related topic	HausdorffSpaceNotCompletelyHausdorff
Related topic	T1Space