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).

Not to be confused with Hilbert curve.

. . . Hilbert cube . . .

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 infinitedimension, where the lengths of the edges in each orthogonal direction form the sequence

{1/n}nN{displaystyle lbrace 1/nrbrace _{nin 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

{an}{displaystyle lbrace a_{n}rbrace }

with

0an1/n{displaystyle 0leq a_{n}leq 1/n}

, then a homeomorphism to the infinite dimensional unit cube is given by

h(a)n=nan{displaystyle h(a)_{n}=ncdot a_{n}}

.

. . . Hilbert cube . . .

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. . . Hilbert cube . . .