Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).

The Hilbert cube is best defined as the topological product of the intervals [0, 1/n] 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 ${\displaystyle \lbrace 1/n\rbrace _{n\in \mathbb {N} }}$.

The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval [0, 1]. 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 ${\displaystyle \lbrace a_{n}\rbrace }$ with ${\displaystyle 0\leq a_{n}\leq 1/n}$, then a homeomorphism to the infinite dimensional unit cube is given by ${\displaystyle h(a)_{n}=n\cdot a_{n}}$.

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