Kolmogoroff room

In topology and related areas of mathematics , a Kolmogoroff space (named after the mathematician Andrei Nikolajewitsch Kolmogorow ), also called T ₀ space , is a topological space in which there are no two different points that are topologically indistinguishable . Clearly speaking, Kolmogoroff spaces never contain several points in the same place , while the general definition of a topological space allows this. The property of being a Kolmogoroff space is also called the axiom and is one of the common axioms of separation . ${\displaystyle T_{0}}$

Topological distinctness

To define, we first introduce the concept of topological distinctness . In a topological space two points hot and topologically indistinguishable , if one of the following equivalent conditions is satisfied: ${\displaystyle T_{0}}$ ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle y}$

• ${\displaystyle x}$and have the same environments , i. H. every open set contains if and only if it contains. ${\displaystyle y}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle y}$
• ${\displaystyle x}$is an element of the conclusion of and belongs to the conclusion of . ${\displaystyle \left\{y\right\}}$ ${\displaystyle y}$ ${\displaystyle \left\{x\right\}}$
• ${\displaystyle x}$and have the same degrees. ${\displaystyle y}$

Otherwise they are called and topologically distinguishable . Topologically indistinguishable points have the same topological properties, i.e. all properties of a point that can be defined by means of the topology of space apply equally to topologically indistinguishable points (because the interchanging of two topologically indistinguishable points is an automorphism , i.e. a homeomorphism in itself) . The topological indistinguishability goes beyond this property, however: the inequality cannot be determined even on the basis of any relationships between the two points that can be expressed by the surroundings, which follows directly from the definition. The presence of additional topologically indistinguishable points does not significantly influence the structure of the room. Topological indistinguishability is preserved under continuous images, the distinguishability under continuous archetypes . ${\displaystyle x}$ ${\displaystyle y}$

Example: In a topological space that is equipped with the indiscreet topology , any two points are topologically indistinguishable.

definition

A topological space is a space if each pair of different points is topologically distinguishable. ${\displaystyle X}$ ${\displaystyle T_{0}}$

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

Another equivalent definition is: is a -space if and only if for any two points in there exists an open set in which contains exactly one of the two points. In contrast to the analogous characterization of a T₁ space, it cannot be predicted which of the two points belongs to the open set. ${\displaystyle X}$ ${\displaystyle T_{0}}$ ${\displaystyle X}$ ${\displaystyle X}$

Kolmogoroff quotient space

Almost all topological spaces studied in mathematics satisfy the axiom . In the event that one nevertheless encounters a topological space that does not fulfill, the space can often be replaced by a space , especially in analysis . This comes in handy in many cases. The following explanations clarify this: With a given set , but where the possibility of varying the topology within certain limits exists, it can be undesirable to force the topology , since non- spaces are often important special cases. So it is important to know both the version with and without of various conditions on topologies . ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle X}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$

Motivating example

To motivate the general ideas, let's start with a familiar example. The space consists of all measurable functions , so that the Lebesgue integral of over is finite. By definition , this room is equipped with a semi-standard . But one would rather get a normalized vector space . The problem is that there are functions that differ from the null function and that have the semi-norm 0 (violation of the definition requirement). 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 normalized space, but which inherits various properties of the semi-normalized space . ${\displaystyle {\mathcal {L}}^{2}(\mathbb {R} )}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$ ${\displaystyle |f(x)|^{2}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \textstyle \|f\|={\sqrt {\int _{\mathbb {R} }|f(x)|^{2}\,\mathrm {d} x}}}$ ${\displaystyle L^{2}(\mathbb {R} )}$

Both the problem definition and the solution are primarily involved in the topologies generated by the standard and semi-standard. A function with half norm 0 is topologically indistinguishable from the null function. The functions identified with one another are precisely the "points" (here functions) that cannot be distinguished topologically in the original semi-normalized space.

definition

Topological indistinguishability is an equivalence relation. No matter what topological space we start with, the quotient space under this equivalence relation is a T ₀ space. This quotient is called the Kolmogoroff quotient of ; it is designated with . If already had -space are and homeomorphic . ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle KQ(X)}$ ${\displaystyle X}$ ${\displaystyle T_{0}}$ ${\displaystyle KQ(X)}$ ${\displaystyle X}$

Two topological spaces are called kolmogoroff equivalent if their Kolmogoroff quotients are homeomorphic. The interesting thing about Kolmogoroff equivalence is that many properties of topological spaces are preserved under this equivalence, i.e. for two kolmogoroff-equivalent spaces neither or both have such a property. On the other hand, the -axiom follows from various other properties of topological spaces , that is, if a space fulfills such a property, then it is a -space. There are only a few exceptions, for example the property of being an indiscreet room. Often the situation is even more comfortable, because many mathematical structures in topological spaces are transferred from to and vice versa. This means that if you have a room without , you can use the Kolmogoroff quotient to construct a room with the same structure and properties. ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle X}$ ${\displaystyle KQ(X)}$ ${\displaystyle T_{0}}$ ${\displaystyle KQ(X)}$ ${\displaystyle T_{0}}$

The example (see Lp space ) can serve as a demonstration of this possibility. From a topological point of view, the semi-normalized space we started with has many additional structures. Such is a vector space with a semi-norm. This defines a semimetric and a uniform structure compatible with the topology . This structure has other properties. The seminorm thus fulfills the parallelogram equation and the uniform structure is complete . The Kolmogoroff quotient, also designated with , retains these properties. is also a complete, semi-normalized space, whose semi-norm satisfies the parallelogram equation. But we even get a little more, because the space is a T ₀ space. Since a semi-normalized space is a normalized space if and only if the underlying topology satisfies T ₀ , it is a completely normalized space whose norm satisfies the parallelogram equation. Such rooms are called Hilbert rooms . We are dealing here with an example that is studied in both mathematics and physics , especially quantum mechanics . ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle L^{2}(\mathbb {R} )}$

Remove T ₀

If you examine the historical development, you will find that although the norm was defined first, the weaker semi-norm was introduced later, i.e. a non- variant of a norm. It is generally possible to introduce such non- versions for properties as well as structures for topological spaces. Let's start with the property of a topological space to be a Hausdorff space . Another property of a topological space can be defined by saying that the space fulfills this property if and only if the Kolmogoroff quotient is a Hausdorff space. This is a reasonable definition, even if it is less well known. Such a space is called a pre-regular space . (The pre-regularity can also be defined directly within the space: any two topologically distinguishable points are separated by surroundings.) Let us now take a structure that can be placed on a topological space, such as a metric . We can put a new structure on topological space by defining on a metric. Here, too, we get a familiar structure, namely a pseudometric . (This allows different points with the distance zero.) ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle X}$ ${\displaystyle KQ(X)}$ ${\displaystyle KQ(X)}$

This provides a natural way of removing the property from the property or structure requirements. In general, it is easier to examine spaces that satisfy, but on the other hand it can also be useful to include spaces without T _{0 in} order to be able to talk directly about representatives of the Kolmogoroff quotient as points. Depending on requirements, the property can be added or removed with the help of the Kolmogoroff quotient. ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{0}}$

Categorical properties

The Kolmogoroff quotient is a covariant , full , essentially surjective functor from the category Top of topological spaces to the category Top₀ of Kolmogoroff spaces.

The Kolmogoroff quotient is a left adjunction of the canonical embedding of Top₀ in Top .

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .