In topology and related branches of mathematics, a topological space X is a T<sub>0</sub> space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other. In a T<sub>0</sub> space, all points are topologically distinguishable.
This condition, called the T<sub>0</sub> condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T<sub>0</sub> spaces. In particular, all T<sub>1</sub> spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T<sub>0</sub> spaces. This includes all T<sub>2</sub> (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods. In another direction, every sober space (which may not be T<sub>1</sub>) is T<sub>0</sub>; this includes the underlying topological space of any scheme. Given any topological space one can construct a T<sub>0</sub> space by identifying topologically indistinguishable points.
T<sub>0</sub> spaces that are not T<sub>1</sub> spaces are exactly those spaces for which the specialization preorder is a nontrivial partial order. Such spaces naturally occur in computer science, specifically in denotational semantics.
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: