  We analyze topological properties of the one-dimensional Bose-Hubbard model
with a quasiperiodic superlattice potential. This system can be realized in
interacting ultracold bosons in optical lattice in the presence of an
incommensurate superlattice potential. We first analyze the quasiperiodic
superlattice made by the cosine function, which we call Harper-like
Bose-Hubbard model. We compute the Chern number and observe a gap-closing
behavior as the interaction strength $U$ is changed. Also, we discuss the
bulk-edge correspondence in our system. Furthermore, we explore the phase
diagram as a function of $U$ and a continuous deformation parameter $\beta$
between the Harper-like model and another important quasiperiodic lattice, the
Fibonacci model. We numerically confirm that the incommensurate charge density
wave (ICDW) phase is topologically non-trivial and it is topologically
equivalent in the whole ICDW region.
