Let H be a separable and infinite-dimensional Hilbert space. Is every closed subset of H homeomorphic to some closed and bounded subset of H?
It is an old result of Klee saying that the infinite-dimensional Hilbert space is homeomorphic with both its unit sphere and its closed unit ball. See e.g. http://www.ams.org/journals/bull/1961-67-03/S0002-9904-1961-10589-2/S0002-9904-1961-10589-2.pdf , and the references therein.