Fine topology and the fact that superharmonic functions are finely continuous have during the last half century lead to studying p-harmonic functions and the Dirichlet problem on finely open sets in Euclidean spaces. This required Sobolev type spaces and a notion of gradient on such non-open sets. Defining those notions is not quite trivial and has been solved in different ways by various authors. In this talk we shall see how modern calculus based on upper gradients on metric spaces naturally leads to fine topology, thus bringing new light on the classical theory, even in Euclidean spaces. In particular, we discuss the Dirichlet problem and p-harmonic functions on bad sets, and compare upper gradients with respect to different underlying spaces. We characterize sets, for which these notions are non-trivial. Fine potential theory and the above results are discussed in the setting of metric spaces and illustrated by examples.
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