regular space
Definition 1.
A topological space^{} is a regular space^{} if it is both a ${T}_{\mathrm{0}}$ space (http://planetmath.org/T0Space) and a ${T}_{\mathrm{3}}$ space (http://planetmath.org/T3Space).
Example. Consider the set $\mathbb{R}$ with the topology^{} $\sigma $ generated by the basis
$$\beta =\{U=V-C:V\text{is open with the standard topology and}C\text{is (infinite) numerable}\}.$$ |
Since $\mathbb{Q}$ is numerable and $\mathbb{R}$ open, the set of irrational numbers $\mathbb{R}-\mathbb{Q}$ is open and therefore $\mathbb{Q}$ is closed. It can be shown that $\mathbb{R}-\mathbb{Q}$ is an open set with this topology and $\mathbb{Q}$ is closed.
Take any irrational number $x$. Any open set $V$ containing all $\mathbb{Q}$ must contain also $x$, so the regular space property cannot be satisfied. Therefore, $(\mathbb{R},\sigma )$ 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 |