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Concept
Hilbert cube
Summary
In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).
The Hilbert cube is best defined as the topological product of the intervals for That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence
The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension. Some authors use the term "Hilbert cube" to mean this Cartesian product instead of the product of the .
If a point in the Hilbert cube is specified by a sequence with then a homeomorphism to the infinite dimensional unit cube is given by
It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (that is, a Hilbert space with a countably infinite Hilbert basis).
For these purposes, it is best not to think of it as a product of copies of but instead as
as stated above, for topological properties, this makes no difference.
That is, an element of the Hilbert cube is an infinite sequence
that satisfies
Any such sequence belongs to the Hilbert space so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.
As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem.
The compactness of the Hilbert cube can also be proved without the axiom of choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube.
In no point has a compact neighbourhood (thus, is not locally compact). One might expect that all of the compact subsets of are finite-dimensional.
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