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In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. These conditions are examples of separation axioms.
Tychonoff spaces are named after Andrey Tychonoff, whose Russian name (Тихонов) is also sometimes transliterated as "Tychonov", "Tikhonov", "Tihonov", or "Tichonov".
Suppose that X is a topological space.
X is a completely regular space iff, given any closed set F and any point x that does not belong to F, there is a continuous function f from X to the real line R such that f(x) = 0 and f(y) = 1 for every y in F. In fancier terms, this condition says that x and F can be separated by a function.
X is a Tychonoff space, or T3½ space, or Tπ space, or completely T3 space if and only if it is both completely regular and HausdorffIn topology and related branches of mathematics, Hausdorff spaces and preregular spaces are kinds of topological spaces. The conditions imposed are the most significant separation axioms. Definitions Suppose that X is a topological space. X is a Hausdorff).
Note that some mathematical literature uses different definitions for the term "completely regular" and the terms involving "T". The definitions that we have given here are the ones usually used today; however, some authors switch the meanings of the two kinds of terms, or use all terms synonymously for only one condition. In Wikipedia, we will use the terms "completely regular" and "Tychonoff" freely, but we'll avoid the less clear "T" terms. In other literature, you should take care to find out which definitions the author is using. (The phrase "completely regular Hausdorff", however, is unambiguous, and always means a Tychonoff space.) For more on this issue, see History of the separation axioms .
Completely regular spaces and Tychonoff spaces are related through the notion of Kolmogorov equivalence. A topological space is Tychonoff iff it's both completely regular and T0In topology and related branches of mathematics, the T spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. The T condition is one of the separation axioms. Topological distinguishability To define T spaces, we first define t. On the other hand, a space is completely regular iff its Kolmogorov quotient is Tychonoff.
Almost every topological space studied in mathematical analysisAnalysis is that branch of mathematics which deals with the real numbers and complex numbers and their functions. It has its beginnings in the rigorous formulation of calculus and studies concepts such as continuity, integration and differentiability in g is Tychonoff, or at least completely regular. For example, the real line is Tychonoff under the standard Euclidean topologyEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers. Other examples include: