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. 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.
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 that contains one of these points and not the other. More precisely the topological space X is Kolmogorov or T 0 {\displaystyle \mathbf {T} _{0}} if and only if:
If a , b ∈ X {\displaystyle a,b\in X} and a ≠ b {\displaystyle a\neq b} , there exists an open set O such that either ( a ∈ O ) ∧ ( b ∉ O ) {\displaystyle (a\in O)\wedge (b\notin O)} or ( a ∉ O ) ∧ ( b ∈ O ) {\displaystyle (a\notin O)\wedge (b\in O)} . 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,
separated ⇒ topologically distinguishable ⇒ distinct 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 also 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 counter examples Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces, T1 spaces and sober spaces are T0.
Spaces that 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 ( ∫ R | f ( x ) | 2 d x ) 1 2 < ∞ {\displaystyle \left(\int _{\mathbb {R} }|f(x)|^{2}\,dx\right)^{\frac {1}{2}}<\infty } . Two functions which are equal almost everywhere are indistinguishable. See also below.
Spaces that are T0 but not T1 The Zariski topology on Spec(R), the prime spectrum of a commutative ring R, is always T0 but generally not T1. The non-closed points correspond to prime ideals which are not maximal. They are important to the understanding of schemes. The particular point topology on any set with at least two elements is T0 but not T1 since the particular point is not closed (its closure is the whole space). An important special case is the Sierpiński space which is the particular point topology on the set {0,1}. The excluded point topology on any set with at least two elements is T0 but not T1. The only closed point is the excluded point. The Alexandrov topology on a partially ordered set is T0 but will not be T1 unless the order is discrete (agrees with equality). Every finite T0 space is of this type. This also includes the particular point and excluded point topologies as special cases. The right order topology on a totally ordered set is a related example. The overlapping interval topology is similar to the particular point topology since every non-empty open set includes 0. Quite generally, a topological space X will be T0 if and only if the specialization preorder on X is a partial order. However, X will be T1 if and only if the order is discrete (i.e. agrees with equality). So a space will be T0 but not T1 if and only if the specialization preorder on X is a non-discrete partial order.
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