# Hilbert parallelotope

The Hilbert parallelotope $I^{\omega}$ is a closed subset of the Hilbert space $\mathbb{R}\ell^{2}$ (The symbol ’$\mathbb{R}$’ has been prefixed to indicate that the field of scalars is $\mathbb{R}$.) defined as

 $I^{\omega}=\{(a_{0},a_{1},a_{2},\ldots)\mid 0\leq a_{i}\leq 1/(i+1)\}$

As a topological space, $I^{\omega}$ is homeomorphic to the product of a countably infinite number of copies of the closed interval $[0,1]$. By Tychonoff’s theorem, this product is compact, so the Hilbert parallelotope is a compact subset of Hilbert space. This fact also explains the notation $I^{\omega}$.

The Hilbert parallelotope enjoys a remarkable universality property — every second countable metric space is homeomorphic to a subset of the Hilbert parallelotope. Since second countability is hereditary, the converse is also true — every subset of the Hilbert parallelotope is a second countable metric space.

Title Hilbert parallelotope HilbertParallelotope 2013-03-22 14:38:32 2013-03-22 14:38:32 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 46C05 Hilbert cube