PlanetMath: regular space

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regular space

(Definition)

Definition 1 A topological space is a regular space if it is both a space and a space.

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

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

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 . Any open set containing all $\mathbbmss{Q}$ must contain also , 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

is exchanged. That is,

is a stronger property than regular.

Bibliography

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.

"regular space" is owned by drini. [ full author list (2) | owner history (1) ]

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Other names:

regular

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Cross-references: references, separation axioms, property, contain, open set, closed, irrational numbers, open, basis, generated by, topological space
There are 10 references to this entry.

This is version 8 of regular space, born on 2002-02-08, modified 2005-03-26.
Object id is 1863, canonical name is RegularSpace.
Accessed 5068 times total.

Classification:

AMS MSC:

54D10 (General topology :: Fairly general properties :: Lower separation axioms )

Pending Errata and Addenda

None.[ View all 1 ]

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