  In the absence of any symmetry constraints we address universal properties of
the boundary charge $Q_B$ for a wide class of nearest-neighbor tight-binding
models in one dimension with one orbital per site but generic modulations of
on-site potentials and hoppings. We provide a precise formulation of the
bulk-boundary correspondence relating the boundary charge of a single band
uniquely to the Zak phase evaluated in a particular gauge. We reveal the
topological nature of $Q_B$ by proving the quantization of a topological index
$eI=\Delta Q_B - \bar{\rho}$, where $\Delta Q_B$ is the change of $Q_B$ when
shifting the lattice by one site towards a boundary and $\bar{\rho}$ is the
average charge per site. For a single band we find this index to be given by
the winding number of the fundamental phase difference of the Bloch wave
function between the two lattice sites defining the boundary of a half-infinite
system. For a given chemical potential we establish a central topological
constraint $I\in\{-1,0\}$ related only to charge conservation of particles and
holes. Our results are shown to be stable against disorder and we propose
generalizations to multi-channel and interacting systems.
