The Cartesian product of a countable infinity of copies of the interval .
It can be denoted
or
, where
and
are the first infinite cardinal and ordinal, respectively.
It is homeomorphic to the product space of any countable infinity of closed bounded
positive-length intervals.
According to another interesting description (Cullen 1968, pp. 164-165), the Hilbert cube can be identified up to homeomorphisms with the metric
space formed by all sequences of real numbers such that
for all
, where the metric is defined as
It is then a subspace of the metric space
called a Hilbert space which is formed by all real
sequences
such that the series
converges.
The Hilbert cube can be used to characterize classes of topological spaces.
1. A topological space that is second countable and T4 is homeomorphic to a subspace of the Hilbert cube.
2. A topological space that is separable and metrizable is homeomorphic to a subspace of the Hilbert cube.
Other statements of this kind are the Tychonoff theorem and Urysohn's metrization theorem.