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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 nontrivial 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 behavior of the system in the weak and strong scattering regimes and observe drastically different shapes of the quasimomentum spectrum.