  A panel structure on a topological space is just a locally finite family of
closed subspaces. A space together with a panel structure is called a space
with faces. In this paper, we define the notion of polyhedral product over a
space with faces. This notion provides a unifying viewpoint on the
constructions of polyhedral products and generalized moment-angle complexes in
various settings. We compute the stable decomposition of these spaces and use
it to study their cohomology ring structures by the partial diagonal maps.
Besides, we can compute the equivariant cohomology ring of the moment-angle
complex over a space with faces with respect to the canonical torus action. The
calculation leads to a notion of topological face ring of a space with faces,
which generalizes the classical notion of face ring of a simplicial complex. We
will see that many known results in the study of polyhedral products and
moment-angle complexes can be reinterpreted from our general theorems on the
polyhedral product over a space with faces. Moreover, we can derive some new
results via our approach in some settings.
