Editing Hilbert cube (section)

==Definition==

The Hilbert cube is best defined as the [[topological product]] of the [[Interval (mathematics)|intervals]] <math>[0, 1/n]</math> for <math>n = 1, 2, 3, 4, \ldots.</math> That is, it is a [[cuboid]] of [[countably infinite]] [[dimension]], where the lengths of the edges in each orthogonal direction form the sequence <math>\left( 1/n \right)_{n \in \N}.</math>

The Hilbert cube is [[homeomorphism|homeomorphic]] to the product of [[countably infinite]]ly many copies of the [[unit interval]] <math>[0, 1].</math>  In other words, it is topologically indistinguishable from the [[unit cube]] of countably infinite dimension. Some authors use the term "Hilbert cube" to mean this Cartesian product instead of the product of the <math>\left[0, \tfrac{1}{n}\right]</math>.{{sfn|Friedman|1981|p=221}}

If a point in the Hilbert cube is specified by a sequence <math>\left( a_n \right)_{n \in \N}</math> with <math>0 \leq a_n \leq 1/n,</math> then a homeomorphism to the infinite dimensional unit cube is given by <math>h(a)_n = n \cdot a_n.</math>