nLab Hilbert cube




The Hilbert cube is the product:

n[0,1n] n[1n,1n]\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 QQ or I ωI^\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 QQ

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

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

This plays an essential role in the Chapman complement theorem.


Let QQ be the Hilbert cube.


  • 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,

Last revised on February 8, 2023 at 16:54:58. See the history of this page for a list of all contributions to it.