Relationships To Other Separation Axioms
If a normal space is R0, then it is in fact completely regular. Thus, anything from "normal R0" to "normal completely regular" is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.
Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
Read more about this topic: Normal Space
Famous quotes containing the words relationships to, separation and/or axioms:
“With only one life to live we can’t afford to live it only for itself. Somehow we must each for himself, find the way in which we can make our individual lives fit into the pattern of all the lives which surround it. We must establish our own relationships to the whole. And each must do it in his own way, using his own talents, relying on his own integrity and strength, climbing his own road to his own summit.”
—Hortense Odlum (1892–?)
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—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)