Hilbert parallelotope

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

$$I^{\omega }=\{(a_0,a_1,a_2,\mathrm{\dots })\mid 0\le a_i\le 1/(i+1)\}$$

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.