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:
“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 (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“The normal present connects the past and the future through limitation. Contiguity results, crystallization by means of solidification. There also exists, however, a spiritual present that identifies past and future through dissolution, and this mixture is the element, the atmosphere of the poet.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)