**Examples and Counterexamples**

Almost every topological space studied in mathematical analysis is Tychonoff, or at least completely regular. For example, the real line is Tychonoff under the standard Euclidean topology. Other examples include:

- Every metric space is Tychonoff; every pseudometric space is completely regular.
- Every locally compact regular space is completely regular, and therefore every locally compact Hausdorff space is Tychonoff.
- In particular, every topological manifold is Tychonoff.
- Every totally ordered set with the order topology is Tychonoff.
- Every topological group is completely regular.
- Generalising both the metric spaces and the topological groups, every uniform space is completely regular. The converse is also true: every completely regular space is uniformisable.
- Every CW complex is Tychonoff.
- Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff.
- The Niemytzki plane is an example of a Tychonoff space which is not normal.

Read more about this topic: Tychonoff Space

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