  We discuss the relation between particle number conservation and topological
phases. In four spatial dimensions, we find that systems belonging to different
topological phases in the presence of a U(1) charge conservation can be
connected adiabatically, i.e., without closing the gap, upon intermediately
breaking this local symmetry by a superconducting term. The time reversal
preserving topological insulator states in 2D and 3D which can be obtained from
the 4D parent state by dimensional reduction inherit this protection by charge
conservation. Hence, all topological insulators can be adiabatically connected
to a trivial insulating state without breaking time reversal symmetry, provided
an intermediate superconducting term is allowed during the adiabatic
deformation. Conversely, in one spatial dimension, non-symmetry-protected
topological phases occur only in systems that break U(1) charge conservation.
These results can intuitively be understood by considering a natural embedding
of the classifying spaces of charge conserving Hamiltonians into the
corresponding Bogoliubov de Gennes classes.
