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  "body": "\n## Definition\n\nLet $X$ be a set.\nA *parametrization* of $X$ is a map $p \\colon U \\to X$, where $U$ is an open subset of $\\R^n$ for some $n \\ge 0$.\nA *diffeology* on $X$ is a set $\\mathcal{D}$ of parametrizations of $X$, called *plots*, with these properties.\n\n1. Covering: every constant parametrization $U \\to X$ is a plot.\n2. Locality: if a parametrization $p \\colon U \\to X$ is such that each $u \\in U$ has an open neighbourhood $V \\subseteq U$ with $p|_V \\in \\mathcal{D}$, then $p \\in \\mathcal{D}$.\n3. Smooth compatibility: let $p \\colon U \\to X$ be a plot and let $V$ be an open subset of some $\\R^m$. For every [[smooth map|0V33BNKZ9S]] $f \\colon V \\to U$, the map $p \\circ f$ is a plot.\n\nA *diffeological space* is a set with a diffeology.\nA map $F \\colon X \\to Y$ between diffeological spaces is *smooth* if $F \\circ p$ is a plot of $Y$ for every plot $p$ of $X$.\n\n## Remarks\n\nExample: let $M$ be a [[smooth manifold|F45VQ47GPV]] and let the plots be all smooth maps $p \\colon U \\to M$.\nConstant maps are smooth, smoothness is a local property, and composites of smooth maps are smooth, so the three clauses hold.\nA map between manifolds is smooth in the usual sense exactly when it is smooth for these diffeologies.\n\nNon-example: on $X = \\R$, let the parametrizations be the real-[[analytic|081WKEDK6C]] maps $U \\to \\R$.\nClauses 1 and 2 hold, but clause 3 fails: the identity of $\\R$ is analytic, while its composite with the smooth non-analytic map $f(t) = e^{-1/t^2}$ for $t \\ne 0$, $f(0) = 0$, is not analytic.\n\nEvery subset and every quotient of a diffeological space carries a natural diffeology.\nDiffeological spaces with smooth maps form a [[cartesian closed category|BFBNQ36QS8]], so the set of smooth maps between two diffeological spaces is again a diffeological space.\n",
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