  We introduce non-Hermitian plasmonic waveguide-cavity systems with
topological edge states (TESs) at singular points. The compound unit cells of
the structures consist of metal-dielectric-metal (MDM) stub resonators
side-coupled to an MDM waveguide. We show that we can realize both a TES and an
exceptional point at the same frequency when a proper amount of loss is
introduced into a finite three-unit-cell structure. We also show that the
finite structure can exhibit both a TES and a spectral singularity when a
proper amount of gain is introduced into the structure. In addition, we show
that we can simultaneously realize a unidirectional spectral singularity and a
TES when proper amounts of loss and gain are introduced into the structure. We
finally show that this singularity leads to extremely high sensitivity of the
reflected light intensity to variations of the refractive index of the active
materials in the structure. TESs at singular points could potentially
contribute to the development of singularity-based plasmonic devices with
enhanced performance.
