Kolmogorov space

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In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms. Its intuitive meaning is that the points of X are topologically distinguishable. These spaces are named after Andrey Kolmogorov.

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Definition

A T0 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 which contains one of these points and not the other.

Note that topologically distinguishable points are automatically distinct. On the other hand, if the singleton sets {x} and {y} are separated, then the points x and y must be topologically distinguishable. That is,

The property of being topologically distinguishable is, in general, stronger than being distinct but weaker than being separated. In a T0 space, the second arrow above reverses; points are distinct if and only if they are distinguishable. This is how the T0 axiom fits in with the rest of the separation axioms.

Examples and nonexamples

Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces and T1 spaces are T0.

Spaces which are not T0

  • A set with more than one element, with the trivial topology. No points are distinguishable.
  • The set R2 where the open sets are the Cartesian product of an open set in R and R itself, i.e., the product topology of R with the usual topology and R with the trivial topology; points (a,b) and (a,c) are not distinguishable.
  • The space of all measurable functions f from the real line R to the complex plane C such that the Lebesgue integral of |f(x)|2 over the entire real line is finite. Two functions which are equal almost everywhere are indistinguishable. See also below.

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