Cantor space

The Cantor space ( after the German mathematician Georg Cantor ) is a topological space. He is - in addition to the Baire space - of particular importance for the descriptive set theory. It finds applications in the theories of infinite games, and infinite machines. The Cantor space is regarded generally as the space of all sequences in the set. He is homeomorphic to the Cantor set, a subspace of the real numbers, that is, all the topological properties are the same. This article addresses this space from the point of view of descriptive set theory, with about embedding into the real numbers does not matter.

Definition

Let the set of all sequences of values ​​or. Considering the discrete topology, so this results in using the product topology, a topology on. with this topological structure is called the Cantor space. As with the discrete topology is a compact Polish space, this also countable product is a compact Polish space. A concrete pre See way to show that it is a Polish space is as follows: the topology induced by a metric which is given as follows:

Here, denote the first, in which the consequences and different. It is even an ultrametric. The space is separable, since ultimately becoming sequences form a countable dense subset. The completeness can show analogously to the real numbers, by means of the dyadic development correspond to the real numbers in the interval, just such effects, however, are identified with an infinite number of en -ending consequences ending in infinitely many en consequences.

Properties of the topology

Many properties of the Cantor space are analogous to those of the Baire space, about possible characterizations of continuity and convergence:

A function is then exactly one point continuous if there exists for each one, so that the first digits determined by the first digits of. A sequence converges if and only if there exists for each one, so from th follower the first digits are always the same. This is in contrast to the dyadic development of real numbers, there may be due to the above identification, the bodies in the development of rational dyadic limits be completely different ( 0.1, 0.11, 0.111, ... converges to 1.000 ... ).

Since the Cantor space is ultrametrisierbar, he is totally disconnected and therefore even a Stone space. In addition, he is a perfect Polish space, since it does not contain isolated points.

The Cantor space is universal in the sense of compact Polish spaces that each compact Polish space is the continuous image of the Cantor space is ( set of Alexandroff - Urysohn ).

Various Cantor spaces

Even larger effects on finite sets lead to the same topology. So it does not matter for the topological considerations, if one allows non-binary alphabets as in an application in the theory of automata. Be about a space with the product topology and given. Define now a mapping associated with each follower by a binary word

Replaced. is a homeomorphism, because: If the first digits in the set, so it is in the picture and at least as many. Reversal steadily: Are defined in the first places, so it is in the picture and at least many.

Indeed, every perfect Polish Stone - space is homeomorphic to the Cantor even space ( equivalently: every perfect metrizable Stone space). (see next section for proof )

Finally, it should be a homeomorphism called the Cantor set: The function

Is a homeomorphism onto its image - the Cantor set, the set of real numbers in the closed unit interval whose ternary development contains no s. The topology of the Cantor space was generated by means of this homeomorphism by the metric on the real numbers, which is complete since all the topology inducing metrics are complete in a compact space.

For universality

The Baire space has the special property that every Polish space is the continuous image of this space. This property has the Cantor space is not, after all it is compact, which is why only compact spaces continuous image can be his. However true that every compact Polish space is the continuous image of the Cantor space (these are precisely the compact Hausdorff spaces that satisfy the second axiom of countability, these are metrizable after Metrisierbarkeitssatz of Urysohn and, since they are compact, with respect to each metric completely, as these are precisely the compact metrizable spaces). For Proof: Let a compact metrizable space. Now, construct a tree of open sets, ie for each word a closed set of natural numbers with the following properties:

• .

To this end, we choose for each point in an enclosed spheres that are sufficiently small to meet the third condition can (about a radius ). Your open cores form an open cover of which is compact as a closed subset of a compact set. Thus, there exists a finite subcover whose cardinality hot, the respective financial statements can now be as for select, the rest are empty. Now let be the space of sequences over the natural numbers, for all indices. is the continuous image of the Cantor space ( the above construction of a homeomorphism for consequences over another finite set equal to a constant, this can be according to a continuous map from to generalize ). The function is uniquely defined by the Intervallschachtelungsprinzip and surjective. Moreover, it is continuous, since convergence is maintained by consequences under this figure. This therefore provides the desired image.

In the case of a space that is also perfect and totally disconnected, can be the disjoint and perfect and select all, which then results in even a homeomorphism.

Similarly, it follows that every perfect Polish space contains the Cantor space, from which the set of Cantor - Bendixson follows that every uncountable Polish space has the cardinality of the continuum. Also, every completely metrizable, perfect room contains the Cantor space.

Boolean algebra

According to the representation theorem for Boolean algebras every Boolean algebra is isomorphic to the Boolean algebra of open and closed sets of a Stone space (totally disconnected, compact Hausdorff space ). The open and closed sets of Cantor space are precisely those that can be (partly) written as a finite union of sets of all sequences having a fixed common prefix, because: The complement of such an amount is apparently again an open set, and as said sets form a basis for the topology with a common prefix, all other open sets must be represented only as an infinite union of such sets whose complement is not then open, since no such base element may be included. Thus, the specified actually all open and closed sets. This Boolean algebra is therefore countable and has no atoms, ie minimum non - zero elements, for every non-empty open and closed set again divided into two such sets. Conversely, suppose given a perfect Stone - space with a countable number of open and closed sets simultaneously. Since a stone space is always zero-dimensional, these quantities provide a base that is therefore countable. It follows from the above characterization that space is homeomorphic to the Cantor space. Now it follows from the representation theorem for Boolean algebras that any two countably infinite Boolean algebra without atoms are isomorphic, since their corresponding Stone space is just always the Cantor space ( would not be the associated Stone space perfectly, so the Boolean algebra possessed atoms ).

Group structure

By means of the component-wise addition is in the Cantor space and a compact, abelian topological group ( products of topological groups are again topological groups ), called Cantor group. This is also considered part of the harmonic analysis, the Walsh functions are characters of this group.