Editing Tychonoff space (section)

==Definitions==
[[File:Separation of a point from a closed set via a continuous function.svg|thumb|300x300px|Separation of a point from a closed set via a continuous function.]]
A topological space <math>X</math> is called '''{{em|completely regular}}''' if points can be [[Separated sets|separated]] from closed sets via (bounded) continuous real-valued functions. In technical terms this means: for any [[closed set]] <math>A \subseteq X</math> and any [[Point (geometry)|point]] <math>x \in X \setminus A,</math> there exists a [[real line|real-valued]] [[continuous function (topology)|continuous function]] <math>f : X \to \R</math> such that <math>f(x)=1</math> and <math>f\vert_{A} = 0.</math> (Equivalently one can choose any two values instead of <math>0</math> and <math>1</math> and even require that <math>f</math> be a bounded function.)

A topological space is called a '''{{em|Tychonoff space}}''' (alternatively: '''{{em|T<sub>3½</sub> space}}''', or {{em|T<sub>π</sub> space}}, or {{em|completely T<sub>3</sub> space}}) 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 [[Kolmogorov space|T<sub>0</sub>]]. On the other hand, a space is completely regular if and only if its [[Kolmogorov quotient]] is Tychonoff.