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  "body": "\n## Definition\n\nLet $X$ be a set, let $\\mathcal{C}$ be a set of maps $c \\colon \\R \\to X$ and let $\\mathcal{F}$ be a set of maps $f \\colon X \\to \\R$.\nThe triple $(X, \\mathcal{C}, \\mathcal{F})$ is a *Frölicher space* if both conditions hold.\n\n1. A map $f \\colon X \\to \\R$ lies in $\\mathcal{F}$ if and only if $f \\circ c$ is a [[smooth function|0V33BNKZ9S]] $\\R \\to \\R$ for every $c \\in \\mathcal{C}$.\n2. A map $c \\colon \\R \\to X$ lies in $\\mathcal{C}$ if and only if $f \\circ c$ is a smooth function $\\R \\to \\R$ for every $f \\in \\mathcal{F}$.\n\nThe elements of $\\mathcal{C}$ are the smooth curves and those of $\\mathcal{F}$ the smooth functions.\nA map $\\Phi \\colon X \\to Y$ of Frölicher spaces is *smooth* if $\\Phi \\circ c \\in \\mathcal{C}_Y$ for every $c \\in \\mathcal{C}_X$; equivalently, $f \\circ \\Phi \\in \\mathcal{F}_X$ for every $f \\in \\mathcal{F}_Y$.\n\n## Remarks\n\nExample: $X = \\R^n$ with $\\mathcal{C}$ the smooth curves and $\\mathcal{F}$ the smooth functions.\nCondition 2 holds, since a curve is smooth when its coordinate functions are.\nCondition 1 is a theorem of Boman: a function on $\\R^n$ is smooth if it is smooth along every smooth curve.\nThe same works for every finite-dimensional [[smooth manifold|F45VQ47GPV]].\n\nNon-example: $X = \\R$, $\\mathcal{C}$ the smooth curves and $\\mathcal{F}$ the polynomial functions.\nCondition 1 fails: $\\sin \\circ c$ is smooth for every smooth curve $c$, but $\\sin$ is not a polynomial.\n\nEvery Frölicher space carries a [[diffeology|7JS0JFR89E]]: its plots are the maps $p$ from open subsets of Euclidean spaces with $f \\circ p$ smooth for every $f \\in \\mathcal{F}$.\nFrölicher spaces and smooth maps form a [[cartesian closed category|BFBNQ36QS8]].\n",
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