Hilbert cube

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

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.

[1] If a point in the Hilbert cube is specified by a sequence

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).

as stated above, for topological properties, this makes no difference.

That is, an element of the Hilbert cube is an infinite sequence

Any such sequence belongs to the Hilbert space

so the Hilbert cube inherits a metric from there.

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.

The Hilbert cube shows that this is not the case.

But the Hilbert cube fails to be a neighbourhood of any point

Any infinite-dimensional convex compact subset of

The Hilbert cube is a convex set, whose span is dense in the whole space, but whose interior is empty.

This situation is impossible in finite dimensions.

The closed tangent cone to the cube at the zero vector is the whole space.

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable.

It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.

In particular, every Gδ-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space.

Conversely, every Polish space is homeomorphic to a Gδ-subset of the Hilbert cube.