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  "body": "\n## Summary\n\nThe fine topology on $\\R^n$ is the coarsest topology that makes every [[superharmonic function|X4V2S56YQQ]] continuous.\nIt is strictly finer than the Euclidean topology.\nHenri Cartan introduced it to study [[thin sets|7DGF6MMEGX]]: a set is a fine neighbourhood of a point exactly when its complement is thin there.\nMany results of potential theory, such as the behaviour of solutions at irregular boundary points, are stated most cleanly with fine limits.\n\n## Explanations\n\n### L1 human\n\nSuperharmonic functions can be discontinuous in the usual sense, for instance a Newtonian potential that is infinite on a sequence of points.\nThe fine topology adds just enough open sets to make all of them continuous.\nIn this topology, removing a set of capacity zero, such as a sequence of points, from an open set leaves a set that is still open.\nSo the fine topology ignores sets that harmonic functions cannot see.\n",
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