  We use an exact solution to the fundamental finite Kronig-Penney model with
arbitrary positions and strengths of scattering sites to show that this iconic
model can possess topologically non-trivial properties. By using free
parameters of the system as extra dimensions we demonstrate the appearance of
topologically protected edge states as well as the emergence of a Hofstadter
butterfly-like quasimomentum spectrum, even in the case of small numbers of
scattering sites. We investigate the behaviour of the system in the weak and
strong scattering regimes and observe drastically different shapes of the
quasimomentum spectrum.
