  We generalise results about isometric group actions on metric spaces and
their fundamental regions to the context of merely continuous group actions. We
will focus on results that yield the relative compactness of a fundamental
region for a cocompact group action, which requires us to replace the condition
of local finiteness by a strictly stronger condition. Finally we will give some
applications of our results, including a necessary condition for homothetic
transformation of a semi-Riemannian manifold to be essential.
