Cantor space


Cantor space

In mathematics, the term Cantor space is sometimes used to denotethe topological abstraction of the classical Cantor set:A topological space is aCantor space if it is homeomorphic to the Cantor set.

The Cantor set itself is of course a Cantor space. Butthe canonical example of a Cantor space is the countably infinite topological product of the discrete 2-point space{0, 1}. This is usually written as 2N or 2ω(where 2 denotes the 2-element set{0,1} with the discrete topology).A point in 2N is aninfinite binary sequence, that is a sequence whichassumes only the values 0 or 1. Given such asequence "a"1, "a"2, "a"3,...one can map it to the real number

:sum_{n=1}^infty frac{2 a_n}{3^n}.

It is not difficult to see that this mapping is ahomeomorphism from 2N onto the Cantor set, and hence that2N is indeed a Cantor space.

A topological characterization of Cantor spaces is givenby Brouwer's theorem::"Any two non-empty compact Hausdorff spaces without isolated points and having countable bases consisting of clopen sets are homeomorphic to each other". (The topological property of having a base consistingof clopen sets is sometimes known as "zero-dimensionality".)This theorem can be restated as: :"A topological space is a Cantor space if and only if it is non-empty, perfect, compact, totally disconnected, and metrizable." It is also equivalent (via Stone's representation theorem for Boolean algebras)to the fact that any two countable atomless
Boolean algebras are isomorphic.

As can be expected from Brouwer's theorem, Cantor spacesappear in several forms. But it is usually easiest to deal with2N, since because ofits special product form, many topological and otherproperties are brought out very explicitly.

For example, it becomes obvious that the cardinality ofany Cantor space is 2^{aleph_0}, that is,the cardinality of the continuum. Also clear is thefact that the product of two(or even any finite or countable number of) Cantor spacesis a Cantor space - an important fact about Cantor spaces.

Using this last fact and the Cantor function, it is easyto construct space-filling curves.

Cantor spaces occur in abundance in real analysis.For example they exist as subspaces in every perfect,
complete metric space. (To see this, note that insuch a space, any non-empty perfect set containstwo disjoint non-empty perfect subsets of arbitrarilysmall diameter, and so one can imitate the constructionof the usual Cantor set.) Also, every uncountable,
separable, completely metrizable space containsCantor spaces as subspaces. This includes most ofthe common type of spaces in real analysis.

Compact metric spaces are also closely related toCantor spaces: A Hausdorff topological space is compactmetrizable if and only if it is a continuous imageof a Cantor space.

ee also

*Cantor cube
*Georg Cantor

References

*cite book | author=Kechris, A. | title= Classical Descriptive Set Theory | publisher=Springer | year=1995 | id = ISBN 0-387-94374-9| edition=Graduate Texts in Mathematics 156


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cantor — may refer to:In general* The Latin word for singer, e.g. the main singer of a cantus * Hazzan , in Judaism, the English name for a professional singer who leads prayer services (Kantor is a frequently noted Jewish patronym) * Cantor (church), an… …   Wikipedia

  • Cantor set — In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883 [Georg Cantor (1883) Über unendliche, lineare Punktmannigfaltigkeiten V [On infinite, linear point manifolds (sets)] , Mathematische Annalen , vol. 21, pages… …   Wikipedia

  • Space-filling curve — 3 iterations of a Peano curve construction, whose limit is a space filling curve. In mathematical analysis, a space filling curve is a curve whose range contains the entire 2 dimensional unit square (or more generally an N dimensional hypercube) …   Wikipedia

  • Cantor cube — In mathematics, a Cantor cube is a topological group of the form {0, 1} A for some index set A . Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given… …   Wikipedia

  • Space (mathematics) — This article is about mathematical structures called spaces. For space as a geometric concept, see Euclidean space. For all other uses, see space (disambiguation). A hierarchy of mathematical spaces: The inner product induces a norm. The norm… …   Wikipedia

  • Space Goofs — Título Space Goofs (Estados Unidos y América Latina), Home to Rent (Reino Unido), Les Zinzins de l espace (Francia), Heim für Aliens, Ein (Alemania). Género Dibujo animado, slapstick Creado por Jean Yves Raimbaud, Philippe Traversat Reparto …   Wikipedia Español

  • Cantor Fitzgerald — Infobox Company company name = Cantor Fitzgerald, L.P. company company type = Private foundation = 1945 location = New York, New York key people = Howard W. Lutnick, Chairman CEO industry = Investment Services products = Financial Services… …   Wikipedia

  • Cantor-Bendixson theorem — noun A theorem which states that a closed set in a Polish space is the disjoint union of a countable set and a perfect set. From the Cantor Bendixson theorem it can be deduced that an uncountable set in must have an uncountable number of limit… …   Wiktionary

  • Perfect space — In mathematics, in the field of topology, perfect spaces are spaces that have no isolated points. In such spaces, every point can be approximated arbitrarily well by other points given any point and any topological neighborhood of the point,… …   Wikipedia

  • Discrete space — In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. Contents 1 Definitions 2 Properties 3 Uses …   Wikipedia