regular space
Definition 1.
A topological space is a regular space
if
it is both a T0 space (http://planetmath.org/T0Space) and a
T3 space (http://planetmath.org/T3Space).
Example.
Consider the set ℝ with the topology σ generated by the basis
β={U=V-C:V is open with the standard topology and C is (infinite) numerable}. |
Since ℚ is numerable and ℝ open, the set of irrational numbers ℝ-ℚ is open and therefore ℚ is closed. It can be shown that ℝ-ℚ is an open set with this topology and ℚ is closed.
Take any irrational number x. Any open set V containing all ℚ must contain also x, so the regular space property cannot be satisfied. Therefore, (ℝ,σ) 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 T3 is exchanged. That is,
T3 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 |