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Concept# Tychonoff space

Summary

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 refers to any completely regular space that is also a Hausdorff space; there exist completely regular spaces that are not Tychonoff (i.e. not Hausdorff).
Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as "Tychonov", "Tikhonov", "Tihonov", "Tichonov", etc. who introduced them in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.
A topological space is called if points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set and any point there exists a real-valued continuous function such that and (Equivalently one can choose any two values instead of and and even demand that be a bounded function.)
A topological space is called a (alternatively: , or , or ) if it is a completely regular Hausdorff space.
Remark. Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff if and only if it's both completely regular and T0. On the other hand, a space is completely regular if and only if its Kolmogorov quotient is Tychonoff.
Across mathematical literature different conventions are applied when it comes to the term "completely regular" and the "T"-Axioms. The definitions in this section are in typical modern usage. Some authors, however, switch the meanings of the two kinds of terms, or use all terms interchangeably. In Wikipedia, the terms "completely regular" and "Tychonoff" are used freely and the "T"-notation is generally avoided. In standard literature, caution is thus advised, to find out which definitions the author is using. For more on this issue, see History of the separation axioms.

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