The Cartesian product of a countable infinity of copies of the interval ![[0,1]](/images/equations/HilbertCube/Inline1.svg)
.
It can be denoted ![[0,1]^(aleph_0)](/images/equations/HilbertCube/Inline2.svg)
or ![[0,1]^omega](/images/equations/HilbertCube/Inline3.svg)
, 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.
