Cantor cube: Difference between revisions

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{{No footnotes|date=January 2020}}

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 the [[discrete topology]]).

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*[[zero-dimensional space|zero-dimensional]];

*AE(0), an [[absolute extensor]] for compact zero-dimensional spaces. (Every map from a closed subset of such a space into a Cantor cube extends to the whole space.)

By a theorem of Schepin, these four properties characterize Cantor cubes; any space satisfying the properties is [[homeomorphic]] to a Cantor cube.

In fact, every AE(0) space is the [[continuous function (topology)|continuous]] [[image (mathematics)|image]] of a Cantor cube, and with some effort one can prove that every [[compact group]] is AE(0). It follows that every zero-dimensional compact group is homeomorphic to a Cantor cube, and every compact group is a continuous image of a Cantor cube.

==References==

*{{cite book | last = Todorcevic | first = Stevo | authorlink=Stevo Todorčević | year = 1997 | title = Topics in Topology | idurl = ISBNhttps://archive.org/details/topicsintopology0000todo | url-access = registration | isbn = 3-540-62611-5}}

*{{springer|author=A.A. Mal'tsev|title=Colon|id=C/c023230}}

==External links==

[[Category:Topological groups]]

{{Sister project links|auto=y|wikt=y}}

[[Category:Topological groups]]

[[pl:Kostka Cantora]]

[[Category:Georg Cantor]]

[[Category:Cubes]]

[[Category:Eponyms in mathematics]]