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

The weak Cartan property for the p-fine topology on metric spaces