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Hilbert Cube


The Cartesian product of a countable infinity of copies of the interval [0,1]. It can be denoted [0,1]^(aleph_0) or [0,1]^omega, where aleph_0 and omega are the first infinite cardinal and ordinal, respectively. It is homeomorphic to the product space of any countable infinity of closed bounded positive-length intervals.

According to another interesting description (Cullen 1968, pp. 164-165), the Hilbert cube can be identified up to homeomorphisms with the metric space formed by all sequences {a_n}_(n=1)^infty of real numbers such that 0<=a_n<=1/n for all n, where the metric is defined as

 g({a_n}_(n=1)^infty,{b_n}_(n=1)^infty)=sqrt(sum_(n=1)^infty(a_n-b_n)^2).

It is then a subspace of the metric space H called a Hilbert space which is formed by all real sequences {a_n}_(n=1)^infty such that the series sum_(n=1)^inftya_n^2 converges.

The Hilbert cube can be used to characterize classes of topological spaces.

1. A topological space that is second countable and T4 is homeomorphic to a subspace of the Hilbert cube.

2. A topological space that is separable and metrizable is homeomorphic to a subspace of the Hilbert cube.

Other statements of this kind are the Tychonoff theorem and Urysohn's metrization theorem.


See also

Tychonoff Theorem, Urysohn's Metrization Theorem

This entry contributed by Margherita Barile

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References

Cullen, H. F. Introduction to General Topology. Boston, MA: Heath, 1968.

Referenced on Wolfram|Alpha

Hilbert Cube

Cite this as:

Barile, Margherita. "Hilbert Cube." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HilbertCube.html

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