  We show that the topological index of a wavefunction, computed in the space
of twisted boundary phases, is preserved under Hilbert space truncation,
provided the truncated state remains normalizable. If truncation affects the
boundary condition of the resulting state, the invariant index may acquire a
different physical interpretation. If the index is symmetry protected, the
truncation should preserve the protecting symmetry. We discuss implications of
this invariance using paradigmatic integer and fractional Chern insulators,
$Z_2$ topological insulators, and Spin-$1$ AKLT and Heisenberg chains, as well
as its relation with the notion of bulk entanglement. As a possible
application, we propose a partial quantum tomography scheme from which the
topological index of a generic multi-component wavefunction can be extracted by
measuring only a small subset of wavefunction components, equivalent to the
measurement of a bulk entanglement topological index.
