TITLE: Why do we care about the Hilbert Cube?
QUESTION [7 upvotes]: The Hilbert Cube is defined to be the countable infinite Cartesian products of the interval $[0,1]$ or anything homeomorphic to $[0,1]$. Why do we care about this object?

REPLY [2 votes]: In addition to Henno's answer and many comments, here is my take, as a geometric topologist. Below is a quote from the introduction to Chapman's book "Lectures on Hilbert cube manifolds" (Q-manifolds, which are analogues of topological manifolds, except that the local structure is modeled on the Hilbert cube):