# Difference between revisions of "Hilbert cube"

From Encyclopedia of Mathematics

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− | The subspace of the [[Hilbert space|Hilbert space]] | + | {{TEX|done}} |

+ | The subspace of the [[Hilbert space|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|compactum]] and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the [[Tikhonov cube|Tikhonov cube]] $I^{\aleph_0}$. It is a [[Universal space|universal space]] in the class of metric spaces with a countable base (Urysohn's metrization theorem). | ||

## Latest revision as of 20:09, 12 April 2014

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=12075

This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article