Hilbert cube

From Encyclopedia of Mathematics
Revision as of 18:57, 7 February 2011 by (talk) (Importing text file)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
How to Cite This Entry:
Hilbert cube. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article