As a topological space, is homeomorphic to the product of a countably infinite number of copies of the closed interval . 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 .
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.
|Date of creation||2013-03-22 14:38:32|
|Last modified on||2013-03-22 14:38:32|
|Last modified by||rspuzio (6075)|