  The topology of typical Chern insulators is rooted in the periodicity of the
system along two directions of real-space. In this article, we depart from this
standard concept and demonstrate that a generic non-Hermitian photonic
waveguide periodic along a single direction of real space can be regarded as a
sub-component of an extended system with a synthetic dimension and with a
nontrivial Chern topology. In particular, we show that the number of bands
below a band-gap of a generic waveguide determines the gap Chern number of the
extended system. It is theoretically and numerically demonstrated that in
real-space the gap Chern number gives the number of gapless Tamm state branches
localized at the system boundary, when its geometry is continuously displaced
by one lattice period. In the non-Hermitian case, the Tamm states connect
different bands in the complex plane.
