Definition
The Hilbert cube is best defined as the topological product of the intervals for n = 1, 2, 3, 4, ... That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence .
The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval . In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension.
If a point in the Hilbert cube is specified by a sequence with, then a homeomorphism to the infinite dimensional unit cube is given by .
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