  We study the hard-core bosons in one-dimensional (1D) interacting topological
bands at different filling factors using exact diagonalization. At the filling
factor $\nu=1$ and in the presence of on-site Hubbard interaction, we find no
sign of the existence of the bosonic topological phase, which is in contrast to
the fermionic case. Instead by studying the momentum distribution and the
condensate fraction we find a superfluid (SF) to Mott-insulator transition
driven by the Hubbard interaction. At the filling factor $\nu=1/3$ and in the
presence of longer-ranged interactions, we identify the bosonic fractional
topological phase (FTP) whose ground-states are characterized by the three-fold
degeneracy and quantized total Berry phase, which is very similar to the
fermionic case. Finally we discuss the reason of the different behaviors of
hard-core bosons at different filling factors by mapping them to spinless
fermions. Our these results can be realized in cold-atom experiments.
