nLab Hilbert cube

Definition

The Hilbert cube is the product:

$$\prod_n [0,\frac{1}{n}]\cong\prod_n [-\frac{1}{n},\frac{1}{n}]$$

It is a compact metrizable space under the sup norm, where the metric topology equals the product topology (as is easily seen). It is variously denoted by $Q$ or $I^\omega$.

It plays a central role in Borsuk's shape theory, and is the basis for the construction of Hilbert cube manifolds. The theory of these were developed by Tom Chapman (mid 1970s) and were used in his proof of the topological invariance of Whitehead torsion.

Pseudo-interior of $Q$

It has an important subspace known as its pseudo-interior. This is the product of the corresponding open intervals,

$$s= \prod_n (-\frac{1}{n},\frac{1}{n}).$$

This plays an essential role in the Chapman complement theorem.

Properties

Let $Q$ be the Hilbert cube.

• A space is second countable (has a countable basis) and $T_4$ (is normal and Hausdorff) if and only if it is homeomorphic to a subspace of $Q$.

• A topological space is Polish if and only if it is homeomorphic to a $G_\delta$-subset of $Q$.

• The Hilbert cube has some counterintuitive properties, such as the fact that it is a homogeneous space (i.e., the group of self-homeomorphisms $Aut(Q)$ acts transitively on $Q$), even though it seems to have a “boundary”. See Halverson and Wright for some explicit constructions.

References

• T.A.Chapman, On Some Applications of Infinite Dimensional Manifolds to the Theory of Shape, Fund. Math. &6 (1972), 181 - 193.

• T.A. Chapman, Lectures on Hilbert Cube Manifolds, CBMS 28, American Mathematical Society, Providence, RI, 1975

• Denise M. Halverson, David G. Wright, The Homogeneous Property of the Hilbert Cube, http://arxiv.org/pdf/1211.1363v1.pdf