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Kolmogorov space
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.
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 if and only if:
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 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.
Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces, T1 spaces and sober spaces are T0.
Commonly studied topological spaces are all T0. Indeed, when mathematicians in many fields, notably analysis, naturally run across non-T0 spaces, they usually replace them with T0 spaces, in a manner to be described below. To motivate the ideas involved, consider a well-known example. The space L2(R) is meant to be 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. This space should become a normed vector space by defining the norm ||f|| to be the square root of that integral. The problem is that this is not really a norm, only a seminorm, because there are functions other than the zero function whose (semi)norms are zero. The standard solution is to define L2(R) to be a set of equivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space; see below.
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Kolmogorov space
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.
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 if and only if:
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 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.
Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces, T1 spaces and sober spaces are T0.
Commonly studied topological spaces are all T0. Indeed, when mathematicians in many fields, notably analysis, naturally run across non-T0 spaces, they usually replace them with T0 spaces, in a manner to be described below. To motivate the ideas involved, consider a well-known example. The space L2(R) is meant to be 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. This space should become a normed vector space by defining the norm ||f|| to be the square root of that integral. The problem is that this is not really a norm, only a seminorm, because there are functions other than the zero function whose (semi)norms are zero. The standard solution is to define L2(R) to be a set of equivalence classes of functions instead of a set of functions directly. This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space. It inherits several convenient properties from the seminormed space; see below.