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In [[mathematics]], a '''Cantor cube''' is a [[topological group]] of the form {0, 1}<sup>''A''</sup> for some index set ''A''. Its algebraic and topological structures are the [[group direct product]] and [[product topology]] from the [[cyclic group of order 2]].
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]]
*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 cube.
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