  We study the topological properties of one dimensional systems undergoing
unitary time evolution. We show that symmetries possessed both by the initial
wavefunction and by the Hamiltonian at all times may not be present in the
time-dependent wavefunction -- a phenomenon which we dub "dynamically-induced
symmetry breaking". This leads to the possibility of a time-varying bulk index
after quenching within non-interacting gapped topological phases. The
consequences are observable experimentally through particle transport
measurements. With reference to the entanglement spectrum, we explain how the
topology of the wavefunction can change out of equilibrium, both for
non-interacting fermions and for symmetry-protected topological phases
protected by antiunitary symmetries.
