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:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)