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DEFINITION Topologically, the Hilbert cube may be defined as the Product of Countably Infinite ly many copies of the Unit Interval {Link without Title} . That is, it is the Cube of countably infinite Dimension . 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's best not to think of it as a product of copies of {Link without Title} , but instead as : × [0,1/2 × [0,1/3] × ยทยทยท; 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. Any such sequence belongs to the Hilbert space l2 , so the Hilbert cube inherits a metric from there. PROPERTIES As a product of Compact Hausdorff Space s, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff Theorem . 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. REFERENCES
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