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Cantor cube

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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 groups 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.)

Topologically, any Cantor cube is:

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

In fact, every AE(0) space is the continuous 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

  • Todorcevic, Stevo (1997). Topics in Topology. ISBN 3-540-62611-5.