Cantor room

The Cantor space (after the German mathematician Georg Cantor ) is a topological space . It is - besides the Baire space - of particular importance for descriptive set theory . It finds applications in the theories of infinite games and infinite automata . The Cantor space is usually seen as the space of all consequences on the crowd . It is homeomorphic to the Cantor set , a subspace of real numbers , i.e. H. all topological properties are the same. This article deals with space from the point of view of descriptive set theory, whereby the embedding in the real numbers does not play a role. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \{0,1\}}$

definition

Let be the set of all sequences of values or . If one looks at the discrete topology , a topology results from this by means of the product topology . with this topological structure is called Cantor space. Since there is a compact Polish space with the discrete topology , this countable product is also a compact Polish space. A more concrete procedure to show that it is a Polish space is as follows: The topology is induced by a metric given as follows: ${\displaystyle {\mathcal {C}}:=\{0,1\}^{\mathbb {N} }}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle \{0,1\}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \{0,1\}}$ ${\displaystyle d}$

${\displaystyle d\colon {\mathcal {C}}\times {\mathcal {C}}\to \mathbb {R} ,(x,y)\mapsto {\frac {1}{2^{n(x,y)}}}}$

Here denote the first place in which the consequences and differ. It is even an ultrametric . The space is separable, since the consequences that ultimately become form a countable, dense subset. The completeness can be shown analogously to the real numbers; by means of the dyadic expansion , the real numbers in the interval correspond to just such sequences, whereby, however, an infinitely many ending sequences are identified with an infinitely many ending sequences. ${\displaystyle n(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle 0}$ ${\displaystyle \left[0,1\right]}$ ${\displaystyle 1}$ ${\displaystyle 0}$

Properties of the topology

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

A function is then exactly one point continuous, if for each one exists, so that the first sites of the first sites of determining. A sequence converges if and only if each one does so from th follower the first places always the same. This is different from the dyadic expansion of real numbers, because of the identification mentioned above, the positions in the expansion for rational, dyadic limit values ​​can be completely different (0.1, 0.11, 0.111, ... converges to 1,000 ...). ${\displaystyle f\colon {\mathcal {C}}\to {\mathcal {C}}}$ ${\displaystyle x}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle m}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle f(x)}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle m}$ ${\displaystyle n}$

Since the Cantor room is ultrametrizable, it is totally incoherent and thus even a stone room . It is also a perfect Polish room as it does not contain any isolated points .

The Cantor space is universal for the compact Polish spaces in the sense that every compact Polish space is a continuous image of the Cantor space ( theorem of Alexandroff-Urysohn ).

Various Cantor rooms

It turns out that Cantor space is homeomorphic to numerous similar or derived structures, which makes it easy to use in descriptive set theory and in automata and game theory: is homeomorphic to for , and . Thus, for example, one can simply speak of projections of quantities without having to switch to a product room. Or relations between elements of the Cantor space can be treated just like simple subsets. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{n}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle {\mathcal {C}}^{\mathbb {N} }}$ ${\displaystyle \{0,1\}\times {\mathcal {C}}}$

Visualization of the adjacent homeomorphism: Above the subspaces of sets with a common prefix in arranged according to prefixes (top level: entire space, second: subspaces with prefix , or , ...), below in . The image of a sub-space has the same color. ${\displaystyle 3^{\mathbb {N} }}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 2^{\mathbb {N} }}$

Sequences over larger finite sets also lead to the same topology. It is therefore irrelevant for topological considerations if, for example, non-binary alphabets are allowed in an application in automata theory . For example, be a room with the product topology and given. Now define a mapping that contains each term in the sequence using a binary word ${\displaystyle {\mathcal {C}}^{\prime }:=\{0,\ldots ,b-1\}^{\mathbb {N} }}$ ${\displaystyle 1<b\in \mathbb {N} }$ ${\displaystyle f\colon {\mathcal {C}}^{\prime }\to {\mathcal {C}}}$ ${\displaystyle k}$ ${\displaystyle u(k)}$

${\displaystyle u(k)=0^{k}1}$for , otherwise ${\displaystyle 0\leq k<b-1}$ ${\displaystyle u(k-1)=0^{k-1}}$

replaced. is a homeomorphism, because: If the first digits are fixed, there are at least as many in the picture. Continuous inversion: If the first digits are specified, there are at least many in the picture . ${\displaystyle f}$ ${\displaystyle \{0,\ldots ,b-1\}^{\mathbb {N} }}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle n}$ ${\displaystyle \lfloor \textstyle {\frac {n}{b}}\rfloor }$

In fact, every perfect Polish Stone room is homeomorphic to the Cantor room (equivalent: every perfect, metrizable Stone room). (see next section on proof)

Finally, a homeomorphism for the Cantor set should be mentioned: The function

${\displaystyle f\colon {\mathcal {C}}\to \left[0,1\right],s\mapsto \sum _{i=0}^{\infty }2\cdot s(i)\cdot 3^{-i-1}}$

is a homeomorphism on your image - the Cantor set, the set of real numbers in the closed unit interval, the ternary expansion of which does not contain any en. The topology of the Cantor space is also generated by means of this homeomorphism by the metric on the real numbers, whereby this is complete, since in a compact space all the metrics that induce the topology are complete. ${\displaystyle 1}$

To universality

The special feature of the Baire room is that every Polish room is a constant image of this room. The Cantor room does not have this property, after all it is compact, which is why only compact rooms can be a constant image of it. However, it is true that every compact Polish space is a continuous image of the Cantor space (these are precisely the compact Hausdorff spaces that satisfy the second axiom of countability ; these are metrizable according to Urysohn's metrisability theorem and, since they are compact, complete with regard to every metric; likewise these are precisely the compact, metrizable rooms). To prove it: be a compact metrizable space. Now construct a tree of open subsets, i.e. for each word a closed set with natural numbers with the following properties: ${\displaystyle X}$ ${\displaystyle w\in \mathbb {N} ^{*}}$ ${\displaystyle C_{w}}$ ${\displaystyle n_{w}}$

• ${\displaystyle C_{\varepsilon }=X}$
• ${\displaystyle C_{wi}\subset C_{w}}$
• ${\displaystyle \operatorname {diam} (C_{w})\to 0}$ For ${\displaystyle |w|\to \infty }$
• ${\displaystyle C_{w}=\bigcup _{i<n_{w}}C_{wi}}$
• ${\displaystyle C_{w}i=\emptyset \Leftrightarrow i\geq n_{w}}$.

To do this, choose for each point in closed spheres that are sufficiently small to be able to meet the third condition (e.g. with a radius ). Their open cores form an open covering of , which is compact as a self-contained subset of a compact. Thus there is a finite partial coverage, the cardinality of which is called, the respective closings can now be selected as for , the rest of them become empty. Let now be the space of the sequences over the natural numbers for which for all indices . is a continuous image of the Cantor space (the above construction of a homeomorphism for sequences over another finite set corresponds to constants , this can be generalized accordingly to a continuous mapping from to ). The function with is clearly defined according to the interval nesting principle and is surjective. In addition, it is continuous because convergence of sequences is preserved under this map. So this provides the desired mapping. ${\displaystyle C_{w}}$ ${\displaystyle \textstyle {\frac {1}{|w|}}}$ ${\displaystyle C_{w}}$ ${\displaystyle n_{w}}$ ${\displaystyle C_{wi}}$ ${\displaystyle i<n_{w}}$ ${\displaystyle C_{wi}}$ ${\displaystyle {\mathcal {C}}^{\prime }}$ ${\displaystyle s}$ ${\displaystyle i}$ ${\displaystyle s(i)<n_{w|i}}$ ${\displaystyle {\mathcal {C}}^{\prime }}$ ${\displaystyle n_{w}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\prime }}$ ${\displaystyle f\colon {\mathcal {C}}^{\prime }\to X}$ ${\displaystyle f(s)\in \textstyle \bigcap _{n\in \mathbb {N} }C_{s|n}}$

In the case of a room that is also perfect and totally disjointed, the disjoint and perfect and all can be chosen, which then even results in a homeomorphism. ${\displaystyle C_{wi}}$ ${\displaystyle n_{w}\geq 2}$

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

Boolean algebra

According to the representation theorem for Boolean algebras , every Boolean algebra is isomorphic to the Boolean algebra of the open and closed sets of a Stone space (totally disconnected, compact Hausdorff space). The open and closed sets of Cantor space are just the ones to be finite union of sets of all sequences with a fixed common prefix ( with can write), because: The complement such an amount is obviously again an open set, and said there Sets with a common prefix form a basis of the topology, all further open sets only have to be represented as an infinite union of those sets whose complement is then not open, since no such basic element can be contained. Thus the specified are actually all open and closed sets. So this Boolean algebra is countable and has no atoms . H. minimal non-zero elements, because every non-empty open and closed set breaks down into two such sets. Conversely, let there be a perfect stone room with a countable number of open and simultaneously closed sets. Since a stone room is always zero-dimensional , these quantities form a basis that can thus be counted. It follows from the above characterization that the space is homeomorphic to the Cantor space. Now it follows from the representation theorem for Boolean algebras that every two countably infinite Boolean algebras without atoms are isomorphic, because their associated Stone space is always the Cantor space (if the associated Stone space were not perfect, Boolean algebra would have atoms ). ${\displaystyle \{s\in {\mathcal {C}}\mid s|_{n}=w\}}$ ${\displaystyle w\in \{0,1\}^{n}}$

Group structure

By means of the component-wise addition in , the Cantor space also becomes a compact, Abelian topological group (products of topological groups are again topological groups), called the Cantor group . This is also considered on the part of the harmonic analysis , the Walsh functions are characters of this group. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$

Individual evidence

1. ^ David Marker: Descriptive Set Theory . 2002, (Lecture notes; PDF; 643 kB).
2. ↑ Paul Alexandroff , Paul Urysohn : Mémoire sur les espaces topologiques compacts (= negotiation of the Koninklijke Akademie van Wetenschappen, Afdeeling Natuurkunde. Section 1: Ingenieurswetenschappen, crystallography, Natuurkunde, Scheikunde, Sterrekunde, Weerkunde en Wiskunde. 14, 1, ZDB -ID 134819 -x ). Uitgave van de Koninklijke Akademie van Wetenschappen, Amsterdam 1929.
3. ↑ Stephen Willard: General Topology. Addison-Wesley, Reading MA et al. 1970, pp. 217, 315.
4. ↑ Eric W. Weisstein : Cantor Set . In: MathWorld (English).
5. ↑ Stephen Willard: General Topology. Addison-Wesley, Reading MA et al. 1970, p. 216.
6. ↑ Alexander S. Kechris : Classical Descriptive Set Theory (= Graduate Texts in Mathematics. Vol. 156). Springer, New York NY et al. 1995, ISBN 3-540-94374-9 , 6.2-6.5.
7. ↑ Nicolas Bourbaki : éléments de mathématique - Topology générale. Chapter ⅠⅩ, p. 114.
8. ↑ Radomir S. Stanković, Jaakko Astola: Remarks on the Development and Recent Results in the Theory of Gibbs Derivatives. In: University of Nis. Facta Universitatis. Electronics and Energetics Series. Vol. 21, No. 3, 2008, pp. 349-364, doi : 10.2298 / FUEE0803349S .