Tychonoff space

Separation axioms
in topological spaces
Kolmogorov classification
T0 (Kolmogorov)
T1 (Fréchet)
T2 (Hausdorff)
T2½(Urysohn)
completely T2 (completely Hausdorff)
T3 (regular Hausdorff)
T3½(Tychonoff)
T4 (normal Hausdorff)
T5 (completely normal
 Hausdorff)
T6 (perfectly normal
 Hausdorff)

In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space is any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff).

Paul Urysohn had used the notion of completely regular space in a 1925 paper[1] without giving it a name. But it was Andrey Tychonoff who introduced the terminology completely regular in 1930.[2]

  1. ^ Urysohn, Paul (1925). "Über die Mächtigkeit der zusammenhängenden Mengen". Mathematische Annalen. 94 (1): 262–295. doi:10.1007/BF01208659. See pages 291 and 292.
  2. ^ Tychonoff, A. (1930). "Über die topologische Erweiterung von Räumen". Mathematische Annalen. 102 (1): 544–561. doi:10.1007/BF01782364.