Examples of Normal Spaces
Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
- All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
- All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
- All compact Hausdorff spaces are normal;
- In particular, the Stone–Čech compactification of a Tychonoff space is normal Hausdorff;
- Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
- All paracompact topological manifolds are perfectly normal Hausdorff. However, there exist non-paracompact manifolds which are not even normal.
- All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
- Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.
Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space that is not regular.
Read more about this topic: Normal Space
Famous quotes containing the words examples of, examples, normal and/or spaces:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (16881744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (16701733)
“What strikes many twin researchers now is not how much identical twins are alike, but rather how different they are, given the same genetic makeup....Multiples dont walk around in lockstep, talking in unison, thinking identical thoughts. The bond for normal twins, whether they are identical or fraternal, is based on how they, as individuals who are keenly aware of the differences between them, learn to relate to one another.”
—Pamela Patrick Novotny (20th century)
“We should read history as little critically as we consider the landscape, and be more interested by the atmospheric tints and various lights and shades which the intervening spaces create than by its groundwork and composition.”
—Henry David Thoreau (18171862)