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Hilbert cube

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The subspace of the Hilbert space $l_2$ consisting of all the points $x=(x_1,x_2,\ldots)$ for which $0\leq x_n\leq(1/2)^n$, $n=1,2,\ldots$. The Hilbert cube is a compactum and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the Tikhonov cube $I^{\aleph_0}$. It is a universal space in the class of metric spaces with a countable base (Urysohn's metrization theorem).


Comments

The topology of the Hilbert cube is studied in the field of infinite-dimensional topology (cf. Infinite-dimensional space). This is a rich and fruitful area of investigation.

See [a1] for an excellent introduction and references.

References

[a1] J. van Mill, "Topology; with an introduction to infinite-dimensional spaces" , North-Holland (1988)
How to Cite This Entry:
Hilbert cube. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hilbert_cube&oldid=31650
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article