Kolmogorov space

In topology and related areas of mathematics a Kolmogorov space (named after the mathematician Andrei Nikolaevich Kolmogorov ), also T0- space is called a topological space in which there are no two different points that are topologically indistinguishable. Intuitively included Kolmogorov - spaces never more points in the same place, while the general definition of a topological space permits. The property to be a Kolmogorov space is also called T0- axiom and is one of the usual separation axioms.

Topological distinguishability

To define T0, we first introduce the concept of topological distinctness. In a topological space X, two points are called x and y are topologically indistinguishable if one of the following equivalent conditions is satisfied:

• X and y have the same environment, i.e., every open set U containing x, if and only if it contains y.
• X is an element of the completion of {y } and y belongs to the completion of { x}.
• X and y have the same qualifications.

Otherwise, x and y are called topologically distinguishable. Topologically indistinguishable points have the same topological properties, ie all the properties of a point can be defined by the topology of the space, apply equally to topologically indistinguishable points (for the interchange of two topologically indistinguishable points is an automorphism, ie a homeomorphism in itself). However, the topological indistinguishability is about this property addition: Also based on any relationship between the two points can be expressed by the environments, the inequality can not determine what follows directly from the definition. The presence of additional topologically indistinguishable points does not affect significantly the structure of space. Topological indistinguishability is preserved under continuous maps, the distinguishability under steady archetypes.

For example, in a topological space, which is equipped with the prying topology, any two points are topologically distinct.

Definition

A topological space X is a T0 space when each pair of different points is topologically distinct.

Topologically distinguishable points are automatically unequal and equal points topologically indistinguishable.

Another equivalent definition is: X is then a T0- space if for any two points in X is an open set in X exists that contains exactly one of the two points. In contrast to the analogous characterization of a T ₁ - space, can not be predicted, which is one of the two points on the open set.

Kolmogorov quotient space

Almost all topological spaces that are studied in mathematics, fulfill the axiom T0. In the event that one still encounters a topological space that does not meet T0, the space can often, especially in the analysis, to be replaced by a T0- space. This proves to be useful in many cases. The following remarks clarify this: For a given set X, but where the possibility of varying the topology within certain limits exist, it may be undesirable to T0 to force the topology to be, as not T0- spaces are often important special cases. So it is important to know of various conditions in each of both topologies the version with and without T0.

Motivating Example

To motivate the general ideas, we begin with a well-known example. The space consists of all measurable functions such that the Lebesgue integral of over is finite. By defining this space is equipped with a semi-norm. But you 'd rather get a normed vector space. The problem is that from the zero function, different functions exist that have the seminorm 0 (violation of Definitheitsforderung ). The standard solution is now to move to a space of equivalence classes. This results in a factor space of the original vector space, and this factor space is a normed space, but inherits a variety of properties of the semi- normed space.

Both the problem and the solution for the topology produced by the standard and half standard involved in the first place. A function with seminorm 0 is topologically indistinguishable from the zero function. The identified together functions are exactly the topologically indistinguishable in the original semi- normed space "points" ( here functions).

Definition

Topological indistinguishability is an equivalence relation. No matter which topological space X we start, the quotient space under this equivalence relation is a T0- space. This quotient is called Kolmogorov quotient of X; it is referred to as KQ (X). If X was a T0- space, may be obtained KQ (X ) and X are homeomorphic.

Two topological spaces hot Kolmogorov - equivalent if their Kolmogorov quotients are homeomorphic. The interesting thing Kolmogorov - equivalence is that we retain many of the properties of topological spaces under this equivalence, possess such a property that is for two Kolmogorov - equivalent spaces none or both. On the other hand, it follows from various other properties of topological spaces, the T0- axiom, that is, if a space satisfies such a property, it is a T0- space. There are only a few exceptions, such as the property of being a indiscrete space. Often the situation is more comfortable, because many mathematical structures on topological spaces transferred from X to KQ (X) and vice versa. This means that if you have an area without T0, one can with the Kolmogorov quotient KQ (X) construct a T0- space with the same structure and properties.

The example (see Lp space) can serve as a demonstration of this possibility. From a topological point of view, the semi- normed space with which we started, many additional structures. So is a vector space with a seminorm. This defines a semi- metric and compatible with the topology of uniform structure. This structure has additional properties. Thus, the semi- norm satisfies the parallelogram law and the uniform structure is complete. The Kolmogorov quotient, also denoted by, retains these properties. is also a full, semi- normed space whose norm satisfies the parallelogram half. We even get a little more, because the space is a T0- space. As a semi- normed space if and only is a normed space, when the underlying topology T0 met, is a complete normed space whose norm satisfies the parallelogram law. Such spaces are called Hilbert spaces. We have here to do it with an example, which is both in mathematics and in physics, especially in quantum mechanics was investigated.

Remove from T0

When one examines the historical development, you will find that although the standard was first defined, and later the weaker semi-norm was introduced, ie a non- T0- variant of a standard. It is generally possible to implement such non- T0 versions both topological features and structures for space. Let's start with the property of a topological space to be a Hausdorff space. One can define another property of a topological space, by saying that the space X then do exactly this property when the Kolmogorov quotient KQ (X ) is a Hausdorff space. This is quite a useful definition, even if it is less well known. Such a space is called präregulären room. ( The Präregularität can also define directly within the space: any two topologically distinguishable points are separated by environments. ) Now let us take a structure that can be placed on a topological space, such as a metric. We can place a new structure on a topological space by defining on KQ ( X) is a metric. Again, we obtain a known structure, namely a pseudo- metric. (This allows different points with the distance to zero. )

This results in a natural way the T0 of the property requirements for a feature or structure to be removed. In general, it is easier to investigate spaces that meet T0, but it can also be useful on the other hand, involve spaces without T0 to talk about Deputy Kolmogorov quotient directly as points can. As required, the T0 property can be added or removed by means of the Kolmogorov - quotient.

Categorical properties

The Kolmogorov quotient is a covariant, full, much surjective functor from the category Top of topological spaces to the category Top ₀ is the Kolmogorov - rooms.

The Kolmogorov quotient is a Linksadjunktion the canonical embedding of Top in Top ₀.