Editing Kolmogorov space (section)

== Definition ==

A '''T<sub>0</sub> space''' is a topological space in which every pair of distinct points is [[topologically distinguishable]]. That is, for any two different points ''x'' and ''y'' there is an [[open set]] that contains one of these points and not the other. More precisely the topological space ''X'' is Kolmogorov or <math>\mathbf T_0</math> if and only if:<ref name=":0" />

:If <math>a,b\in X</math> and <math>a\neq b</math>, there exists an open set ''O'' such that either <math>(a\in O) \wedge (b\notin O)</math> or <math>(a\notin O) \wedge (b\in O)</math>.

Note that topologically distinguishable points are automatically distinct. On the other hand, if the [[singleton set]]s {''x''} and {''y''} are [[separated sets|separated]] then the points ''x'' and ''y'' must be topologically distinguishable. That is,
:''separated'' ⇒ ''topologically distinguishable'' ⇒ ''distinct''
The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T<sub>0</sub> space, the second arrow above also reverses; points are distinct [[if and only if]] they are distinguishable. This is how the T<sub>0</sub> axiom fits in with the rest of the [[separation axiom]]s.