A universal topological group with countable base (Q1086363)

If \(Q\) is the Hilbert cube and \(\Aut Q\) is the group of all homeomorphisms from \(Q\) to itself with the compact-open topology then the author proves the following theorem: Any topological group with countable basis is topologically isomorphic to some subgroup of the group \(\Aut Q\). The consequence is that there exists a separable Banach space \(E\) such that any topological group with countable basis is isomorphic to some subgroup of the group of all linear isometries on \(E\). It holds for \(E=C(Q)\).