Cantor cube


In mathematics, a Cantor cube is a topological group of the form 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.
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.
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 Cantor cube.
In fact, every AE space is the continuous image of a Cantor cube, and with some effort one can prove that every compact group is AE. 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.