Tychonoff space

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A Hausdorff space X is called a Tychonoff space if, for every nonempty closed subset C and every x in the complement of C, there is a continous function f : X -> [0,1] such that f(x) = 0 and f(C) = {1}. Tychonov spaces are also called T3 1/2 spaces, Tπ spaces or completely regular spaces, although these other terms are sometimes used for non-Hausdorff spaces with the above property.

Note that Tychonoff is a Russian name and there are several ways to transliterate it. Alternatives include Tychonov, Tikhonov, Tihonov and Tichonov.

Tychonoff spaces are precisely those topological spaces which can be embedded in a compact Hausdorff space.

Examples of Tychonoff spaces include:

  • All metric spaces.
  • All locally compact Hausdorff spaces, and therefore all manifolds.
  • All linearly ordered topological spaces.
  • All Hausdorff topological groups.
  • All CW complexes.
  • All products of Tychonoff spaces.
  • All subspaces of Tychonoff spaces.