Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below). In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below). 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 { 1 / n } n ∈ N {displaystyle lbrace 1/n brace _{nin mathbb {N} }} . 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 { a n } {displaystyle lbrace a_{n} brace } with 0 ≤ a n ≤ 1 / n {displaystyle 0leq a_{n}leq 1/n} , then a homeomorphism to the infinite dimensional unit cube is given by h ( a ) n = n ⋅ a n {displaystyle h(a)_{n}=ncdot a_{n}} . It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (i.e. a Hilbert space with a countably infinite Hilbert basis).For these purposes, it is best not to think of it as a product of copies of , but instead as as stated above, for topological properties, this makes no difference.That is, an element of the Hilbert cube is an infinite sequence