Tychonoff space

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. 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. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, whose Russian name (Тихонов) is variously rendered as 'Tychonov', 'Tikhonov', 'Tihonov', 'Tichonov' etc. A topological space, X {displaystyle X} , is called completely regular exactly in case points can be separated from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any closed set A ⊆ X {displaystyle Asubseteq X} and any point x ∈ X ∖ A {displaystyle xin Xsetminus A} , then there exists a real-valued continuous function f : X ⟶ R {displaystyle f:Xlongrightarrow mathbb {R} } such that f ( x ) = 1 {displaystyle f(x)=1} and f ∣ A ≡ 0 {displaystyle fmid Aequiv 0} . (Equivalently one can choose any two values instead of 0 {displaystyle 0} and 1 {displaystyle 1} and even demand that f {displaystyle f} be a bounded function.) A topological space, X {displaystyle X} , is furthermore called a Tychonoff space (alternatively: T3½ space, or Tπ space, or completely T3 space) in case 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. 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: Complete regularity and the Tychonoff property are well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies. It follows that: Like all separation axioms, complete regularity is not preserved by taking final topologies. In particular, quotients of completely regular spaces need not be regular. Quotients of Tychonoff spaces need not even be Hausdorff. There are closed quotients of the Moore plane which provide counterexamples.