The Cartesian product of a countable infinity of copies of the interval [0, 1]. It can be denoted [0, 1]^(ℵ_0) or [0, 1]^ω, where ℵ_0 and ω 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, the Hilbert cube can be identified up to homeomorphisms with the metric space formed by all sequences {a_n}_(n = 1)^∞ of real numbers such that 0<=a_n<=1/n for all n, where the metric is defined as g({a_n}_(n = 1)^∞, {b_n}_(n = 1)^∞) = sqrt( sum_(n = 1)^∞ (a_n - b_n)^2).

Tychonoff theorem | Urysohn's metrization theorem