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.