Cantor cube: Difference between revisions

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In [[mathematics]], a '''Cantor cube''' is a [[topological group]] of the form {0, 1}^''A'' for some index set ''A''. If ''A'' is a [[countably infinite set]], the corresponding Cantor cube is a [[Cantor space]]. Cantor cubes are special among [[compact group]]s because every compact group is a continuous image of one, although usually not a homomorphic image. (The literature can be unclear, so for safety, assume all spaces are [[Hausdorff space|Hausdorff]].)

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]]).

In [[mathematics]], a '''Cantor cube''' is a [[topological group]] of the form {0, 1}^''A'' for some index set ''A''. If ''A'' is a [[countably infinite set]], the corresponding Cantor cube is a [[Cantor space]]. Cantor cubes are special among [[compact group]]s because every compact group is a continuous image of one, although usually not a homomorphic image. (The literature can be unclear, so for safety, assume all spaces are [[Hausdorff space|Hausdorff]].)

Topologically, any Cantor cube is:

*[[homogeneous space|homogeneous]];

*[[compact space|compact]];

*[[zero-dimensional space|zero-dimensional]] when ''A'' is finite;

*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==

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

[[Category:Topological groups]]

[[Category:Georg Cantor]]

[[Category:Cubes]]

[[Category:Eponyms in mathematics]]