Relation To Other Separation Axioms
It is an easy exercise to show that any two points which can be separated by a function can be separated by closed neighborhoods. If they can be separated by closed neighborhoods then clearly they can be separated by neighborhoods. It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff.
One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff. In summary we have the following implications:
| Tychonoff (T3½) | regular Hausdorff (T3) | |||||
| completely Hausdorff | Urysohn (T2½) | Hausdorff (T2) | T1 |
One can find counterexamples showing that none of these implications reverse.
Read more about this topic: Completely Hausdorff Space
Famous quotes containing the words relation to, relation, separation and/or axioms:
“Among the most valuable but least appreciated experiences parenthood can provide are the opportunities it offers for exploring, reliving, and resolving one’s own childhood problems in the context of one’s relation to one’s child.”
—Bruno Bettelheim (20th century)
“The difference between objective and subjective extension is one of relation to a context solely.”
—William James (1842–1910)
“... the separation of church and state means separation—absolute and eternal—or it means nothing.”
—Agnes E. Meyer (1887–1970)
“The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)