  First, we generalize the definition of a locally compact topology given by
Paterson and Welch for a sequence of locally compact spaces to the case where
the underlying spaces are $T_1$ and sober. We then consider a certain
semilattice of basic open sets for this topology on the space of all paths on a
graph and impose relations motivated by the definitions of graph C*-algebra in
order to recover the boundary path space of a graph. This is done using
techniques of pointless topology. Finally, we generalize the results to the
case of topological graphs.
