Hilbert cube

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In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in topology. It is named after the German mathematician David Hilbert.

1 Definition
2 The Hilbert cube as a metric space
3 Properties
4 References

[edit] Definition

The Hilbert cube is best defined as the topological product of the intervals [0,1/n] where n = 1,2,3,4... That is, it is a cuboid of countably infinite dimension where the length of each edge has a length 1/n ( $n\in\mathbb{N}$ ) with the extra condition that none of the edges are of equal length.

Topologically, the Hilbert cube is indistinguishable from the product of countably infinitely many copies of the unit interval [0,1]. That is, it is the cube of countably infinite dimension.

[edit] The Hilbert cube as a metric space

It's sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a Hilbert space with countably infinite dimension. For these purposes, it's best not to think of it as a product of copies of [0,1], but instead as

[0,1] × [0,1/2] × [0,1/3] × ···;

as stated above, for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence

(x_n)

that satisfies

0 ≤ x_n ≤ 1/n.

Any such sequence belongs to the Hilbert space ℓ₂, so the Hilbert cube inherits a metric from there.

[edit] Properties

As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem.

Since ℓ₂ is not locally compact, no point has a compact neighbourhood, so one might expect that all of the compact subsets are finite-dimensional. The Hilbert cube shows that this is not the case. But the Hilbert cube fails to be a neighbourhood of any point p because its side becomes smaller and smaller in each dimension, so that an open ball around p of any fixed radius e > 0 must go outside the cube in some dimension.