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).

Definition

The Hilbert cube is best defined a 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 \( \lbrace 1/n \rbrace_{n\in\mathbb{N}}. \)

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 \lbrace a_n \rbrace with 0 \leq a_n \leq 1/n, then a homeomorphism to the infinite dimensional unit cube is given by \( h : a_n \rarr n\cdot a_n. \)
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 is 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. One can show that the topology induced by the metric is the same as the product topology in the above definition.

Properties

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

In ℓ₂, no point has a compact neighbourhood (thus, ℓ₂ is not locally compact). One might expect that all of the compact subsets of ℓ₂ 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.

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

Trivially, every G_δ-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a complete metric space. Conversely, every Polish space is homeomorphic to a G_δ-subset of the Hilbert cube.^[1]

Notes

^ Srivastava, pp. 55

References

Srivastava, Sashi Mohan (1998). A Course on Borel Sets. Graduate Texts in Mathematics. Springer-Verlag. ISBN 978-0-387-98412-4. Retrieved 12-04-08.
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR507446