# 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