Kolmogorov space

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In topology and related branches of mathematics, a topological space X is a T₀ 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.^[1] In a T₀ space, all points are topologically distinguishable.

This condition, called the T₀ condition, is the weakest of the separation axioms. Nearly all topological spaces normally studied in mathematics are T₀ spaces. In particular, all T₁ spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T₀ spaces. This includes all T₂ (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 T₁) is T₀; this includes the underlying topological space of any scheme. Given any topological space one can construct a T₀ space by identifying topologically indistinguishable points.

T₀ spaces that are not T₁ 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.