Tychonoff Space - Examples and Counterexamples

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.

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