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regular space
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(Definition)
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Example. Consider the set
 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
 is numerable and
 open, the set of irrational numbers
 is open and therefore
 is closed. It can be shown that
 is an open set with this topology and
 is closed.
Take any irrational number $x$ . Any open set $V$ containing all
 must contain also $x$ , so the regular space property cannot be satisfied. Therefore,
 is not a regular space.
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.
- 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.
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"regular space" is owned by drini. [ full author list (2) | owner history (1) ]
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Cross-references: stronger, references, separation axioms, property, contain, open set, closed, irrational numbers, open, basis, generated by, topological space
There are 7 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 5900 times total.
Classification:
| AMS MSC: | 54D10 (General topology :: Fairly general properties :: Lower separation axioms ) |
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Pending Errata and Addenda
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