Hilbert cube

The subspace of the Hilbert space $l_2$ consisting of all the points $x=(x_1,x_2,\ldots)$ for which $0\leq x_n\leq(1/2)^n$, $n=1,2,\ldots$. 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 $I^{\aleph_0}$. It is a universal space in the class of metric spaces with a countable base (Urysohn's metrization theorem).

Comments

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.

References

[a1]	J. van Mill, "Topology; with an introduction to infinite-dimensional spaces" , North-Holland (1988)