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:
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“Hesitation increases in relation to risk in equal proportion to age.”
—Ernest Hemingway (1899–1961)
“Like sleep disturbances, some worries at separation can be expected in the second year. If you accept this, then you will avoid reacting to this anxiety as if it’s your fault. A mother who feels guilty will appear anxious to the child, as if to affirm the child’s anxiety. By contrast, a parent who understands that separation anxiety is normal is more likely to react in a way that soothes and reassures the child.”
—Cathy Rindner Tempelsman (20th century)
“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)