The weak Cartan property for the p-fine topology on metric spaces Jana BjornAnders BjornVisa Latvala Primary: 31E05Secondary: 30L9931C4031C4535J9249Q20.capacitycoarsest topologydoublingfine topologyfinely continuousmetric spacep-harmonicPoincare inequalityquasicontinuoussuperharmonicthickthinweak Cartan propertyWiener criterion We study the $p\mspace{1mu}$-fine topology on complete metric spaces e\-quipped with a doubling measure supporting a $p\mspace{1mu}$-Poincar\'e inequality, $1<p<\infty$. We establish a weak Cartan property, which yields characterizations of the $p\mspace{1mu}$-thinness and the $p\mspace{1mu}$-fine continuity, and allows us to show that the $p\mspace{1mu}$-fine topology is the coarsest topology making all $p\mspace{1mu}$-superharmonic functions continuous. Our $p\mspace{1mu}$-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients, and do not rely on a vector-valued differentiable structure. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5527 10.1512/iumj.2015.64.5527 en Indiana Univ. Math. J. 64 (2015) 915 - 941 state-of-the-art mathematics http://iumj.org/access/