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

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.