  We investigate $C^*$-algebras associated with row-finite topological
higher-rank graphs with no source, which are based on product system
$C^*$-algebras. We prove the Cuntz-Krieger uniqueness theorem, and provide the
condition of simplicity and purely infiniteness of our algebras. We give
examples of non-discrete topological higher-rank graphs whose $C^*$-algebras
contain Cuntz's $ax+b$-semigroup $C^*$-algebra over $\mathbb{N}$.
