The subspace of the Hilbert space consisting of all the points for which , . The Hilbert cube is a compactum and is topologically equivalent (homeomorphic) to the Tikhonov product of a countable system of intervals, i.e. to the Tikhonov cube . It is a universal space in the class of metric spaces with a countable base (Urysohn's metrization theorem).
The topology of the Hilbert cube is studied in the field of infinite-dimensional topology (cf. Infinite-dimensional space). This is a rich and fruitful area of investigation.
See [a1] for an excellent introduction and references.
|[a1]||J. van Mill, "Topology; with an introduction to infinite-dimensional spaces" , North-Holland (1988)|
Hilbert cube. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hilbert_cube&oldid=12075