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  "body": "\n## Definition\n\nLet $X$ be a connected, locally connected, [[locally compact|0FE2SB8WVJ]] Hausdorff space.\nLet $\\mathcal{H}$ assign to each open $U \\subseteq X$ a linear subspace $\\mathcal{H}(U)$ of the real [[continuous functions|0X3WGX95FE]] on $U$.\nCall an open set *regular* as in a [[harmonic space|DWN0XEQHHK]].\nThe pair $(X, \\mathcal{H})$ is a *Brelot space* if:\n\n1. $\\mathcal{H}$ is a [[sheaf|TB6TM6R4MD]] of functions.\n2. The regular domains, that is, the connected regular open sets, form a base of the topology of $X$.\n3. Let $U \\subseteq X$ be a domain and $(h_n)$ an increasing sequence in $\\mathcal{H}(U)$. Then $\\lim_n h_n$ either lies in $\\mathcal{H}(U)$ or is $+\\infty$ at every point of $U$.\n\n## Remarks\n\nClause 3 implies the convergence axiom of a harmonic space: a locally bounded limit cannot be identically $+\\infty$ on any component.\nExample: $\\R^n$ with the classical harmonic functions is a Brelot space.\nBalls are regular domains, by the Poisson integral, and clause 3 is [[Harnack's principle|P15K8RKHDV]].\nNon-example: $\\R \\times \\R$ with the solutions of the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\partial_x^2 u$ fails clause 3.\nLet $\\Phi(x, t) = (4\\pi t)^{-1/2} e^{-x^2/(4t)}$ for $t \u003e 0$ and $\\Phi(x, t) = 0$ for $t \\le 0$, the [[heat kernel|KD404MF042]] extended by $0$.\nIt solves the heat equation on the domain $U = \\R^2 \\setminus \\set{(0, 0)}$, and $h_n = n \\Phi$ increases to $+\\infty$ where $t \u003e 0$ and to $0$ where $t \\le 0$.\nThe limit is neither a solution on $U$ nor identically $+\\infty$.\n",
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