Definition
The Hilbert cube is best defined as the topological product of the intervals for n = 1, 2, 3, 4, ... That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence .
The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval . In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension.
If a point in the Hilbert cube is specified by a sequence with, then a homeomorphism to the infinite dimensional unit cube is given by .
Read more about this topic: Hilbert Cube
Famous quotes containing the word definition:
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)