The Weak Cartan Property for the p-fine Topology on Metric Spaces

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p < infinity. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all p-superharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients, and do not rely on a vector-valued differentiable structure.