  Using a prototype model for proximity induced superconductivity on a bilayer
square lattice, we show that interlayer tunneling can drive change in topology
of the Bogoliubov quasiparticle bands. Starting with topologically trivial
superconductors, transitions to a non-trivial $p_x + {\rm i} p_y$ state and
back to another trivial state are discovered. We characterize these phases in
terms of edge-state spectra and Chern indices. We show that these transitions
can also be controlled by experimentally viable control parameters, the
bandwidth of the metallic layer and the gate potential. Insights from our
results on a simple model for proximity induced superconductivity may open up a
new route to discover topological superconductors.
