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–?)
“A separation situation is different for adults than it is for children. When we were very young children, a physical separation was interpreted as a violation of our inalienable rights....As we grew older, the withdrawal of love, whether that meant being misunderstood, mislabeled or slighted, became the separation situation we responded to.”
—Roger Gould (20th century)
““I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.””
—Henry Brooks Adams (1838–1918)