- Cantor space
In

mathematics , the term**Cantor space**is sometimes used to denotethe topological abstraction of the classicalCantor set :Atopological space is aCantor space if it ishomeomorphic to theCantor 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**2**^{N}or**2**^{ω}(where**2**denotes the 2-element set{0,1} with the discrete topology).A point in**2**^{N}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

**2**^{N}onto theCantor set , and hence that**2**^{N}is indeed a Cantor space.A topological characterization of Cantor spaces is givenby Brouwer's theorem::"Any two non-empty compact

Hausdorff space s withoutisolated point s and having countable bases consisting ofclopen set s 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 , andmetrizable ." It is also equivalent (viaStone'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 with

**2**^{N}, 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 constructspace-filling curve s.Cantor spaces occur in abundance in

real analysis .For example they exist as subspaces in every perfect,

completemetric 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 usualCantor 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

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