  We identify a class of dressed atom-photon states formingat the same energy
of the atom at any coupling strength. As a hallmark, their photonic component
is an eigenstate of the bare photonic bath with a vacancy in place of the atom.
The picture accommodates waveguide-QED phenomena where atoms behave as perfect
mirrors, connecting in particular dressed bound states (BS) in the continuum or
BIC with geometrically-confined photonic modes. When applied to photonic
lattices, the framework provides a general criterion to predict dressed BS in
lattices with topological properties by putting them in one-to-one
correspondence with photonic BS. New classes of dressed BS are thus predicted
in the photonic Creutz-ladder and Haldane models. In the latter case, states
with non-zero local photon flux occur, where an atom is dressed by a photon
orbiting around it.
