Hilbert Cube

The Hilbert cube is the topological space formed as the countably infinite product of copies of the closed unit interval [0,1], each shrunk by a diminishing factor so the product carries a well-defined metric and remains compact. It serves as a canonical example of an infinite-dimensional compact space and, remarkably, every separable metric space can be embedded within it, making the Hilbert cube a universal space for that entire class. The space is named after David Hilbert and is studied extensively in infinite-dimensional topology, where its own unusual properties, such as being homeomorphic to some of its own proper subsets, show how infinite-dimensional compact spaces can behave very differently from finite-dimensional ones.