Editing Cantor space (section)

== Characterization ==

A topological characterization of Cantor spaces is given by [[Luitzen Egbertus Jan Brouwer|Brouwer]]'s theorem:<ref>{{citation|first=L. E. J.|last=Brouwer|authorlink=L. E. J. Brouwer|title=On the structure of perfect sets of points|journal=Proc. Koninklijke Akademie van Wetenschappen|volume=12|year=1910|pages=785–794|url=http://www.dwc.knaw.nl/DL/publications/PU00013496.pdf}}.</ref>
{{Math theorem
|name=Brouwer's theorem
|Any two non-empty [[compact space|compact]] [[Hausdorff space]]s without [[isolated point]]s and having countable [[base (topology)|base]]s consisting of [[clopen set]]s are homeomorphic to each other.}}
The topological property of having a base consisting of clopen sets is sometimes known as "[[zero-dimensional|zero-dimensionality]]". Brouwer's theorem can be restated as:
{{Math theorem|A topological space is a Cantor space [[if and only if]] it is non-empty, [[perfect set|perfect]], compact, [[totally disconnected]], and [[metrizable]].}}
This theorem is also equivalent (via [[Stone's representation theorem for Boolean algebras]]) to the fact that any two [[countable atomless Boolean algebra]]s are [[isomorphic]].