  We study, by means of the density-matrix renormalization group (DMRG)
technique, the evolution of the ground state in a one-dimensional topological
insulator, from the non-interacting to the strongly-interacting limit, where
the system can be mapped onto a topological Kondo-insulator model. We focus on
a toy model Hamiltonian (i.e., the interacting "$sp$-ladder" model), which
could be experimentally realized in optical lattices with higher orbitals
loaded with ultra-cold fermionic atoms. Our goal is to shed light on the
emergence of the strongly-interacting ground state and its topological
classification as the Hubbard-$U$ interaction parameter of the model is
increased. Our numerical results show that the ground state can be generically
classified as a symmetry-protected topological phase of the Haldane-type, even
in the non-interacting case $U=0$ where the system can be additionally
classified as a time-reversal $\mathbb{Z}_{2}$-topological insulator, and
evolves adiabatically between the non-interacting and strongly interacting
limits.
