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  "body": "\n## Definition\n\nLet $n \\ge 2$.\nThe *fine topology* on $\\R^n$ is the coarsest [[topology|0CQHFPGJ45]] $\\mathcal{T}$ on $\\R^n$ with the following property.\nFor every open set $U \\subseteq \\R^n$ and every [[superharmonic function|X4V2S56YQQ]] $u$ on $U$, the set $U$ belongs to $\\mathcal{T}$ and $u \\colon U \\to (-\\infty, +\\infty]$ is continuous for $\\mathcal{T}$.\nSets in $\\mathcal{T}$ are called *finely open*, and limits and continuity for $\\mathcal{T}$ are called fine limits and fine continuity.\n\n## Remarks\n\nThe fine topology is [[finer|072J68M6HX]] than the Euclidean topology, since every Euclidean open set is finely open by definition.\nA set $V$ is finely open if and only if, for every $x \\in V$, the complement $\\R^n \\setminus V$ is [[thin|7DGF6MMEGX]] at $x$.\nExample: let $V = \\R^3 \\setminus \\set{k^{-1} e_1 : k \\ge 1}$.\nThe removed set is countable, hence a [[polar set|C95QJ7YH5T]], and a polar set is thin at every point.\nSo $V$ is finely open, but it is not Euclidean open, because every ball about $0 \\in V$ meets the removed points.\nNon-example: a closed half-space $H$ is not a fine neighbourhood of a point $x \\in \\partial H$, since the complementary open half-space is not thin at $x$.\n",
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