  In this paper we study the formation of topological Tamm states at the
interface between a semi-infinite one-dimensional photonic-crystal and a metal.
We show that when the system is topologically non-trivial there is a single
Tamm state in each of the band-gaps, whereas if it is topologically trivial the
band-gaps host no Tamm states. We connect the disappearance of the Tamm states
with a topological transition from a topologically non-trivial system to a
topologically trivial one. This topological transition is driven by the
modification of the dielectric functions in the unit cell. Our interpretation
is further supported by an exact mapping between the solutions of Maxwell's
equations and the existence of a tight-binding representation of those
solutions. We show that the tight-binding representation of the 1D photonic
crystal, based on Maxwell's equations, corresponds to a
Su-Schrieffer-Heeger-type model (SSH-model) for each set of pairs of bands.
Expanding this representation near the band edge we show that the system can be
described by a Dirac-like Hamiltonian. It allows one to characterize the
topology associated with the solution of Maxwell's equations via the winding
number. In addition, for the infinite system, we provide an analytical
expression for the photonic bands from which the band-gaps can be computed.
