Normal Space - Examples of Normal Spaces

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.

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