## Encyclopedia > Hilbert cube

Article Content

# Hilbert cube

In mathematics, the Hilbert cube is a topological space that provides an instructive example of some ideas in topology.

Topologically, the Hilbert cube may be defined as the product of countably infinitely many copies of the unit interval [0,1]. That is, it is the cube of countably infinite dimension. As a product of compact Hausdorff spaces, it is itself a compact Hausdorff space as a result of the Tychonoff theorem.

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] × ···; for topological properties, this makes no difference. That is, an element of the Hilbert cube is an infinite sequence (xn) that satisfies 0 ≤ xn ≤ 1/n. Note that any such sequence belongs to the Hilbert space l2, so the Hilbert cube inherits a metric from there.

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Rock and Roll Hall of Fame ... or groups are qualified to be inducted 25 years after the release of their first record. Inductees 500 Songs that Shaped Rock and Roll External links The Rock and ...