nLab
Hilbert cube

Definition

The Hilbert cube is the product:

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

It is a compact metric space. 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,

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

This plays an essential role in the Chapman complement theorem.

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