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. 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. 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 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. Nearly all topological spaces normally studied in mathematics are T0. In particular, all Hausdorff (T2) spaces and T1 spaces are T0. Examples of topological space typically studied are 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. In general, when dealing with a fixed topology T on a set X, it is helpful if that topology is T0. On the other hand, when X is fixed but T is allowed to vary within certain boundaries, to force T to be T0 may be inconvenient, since non-T0 topologies are often important special cases. Thus, it can be important to understand both T0 and non-T0 versions of the various conditions that can be placed on a topological space.