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 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 the Cantor set, and hence that2^N 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 with2^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 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