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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.
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