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  "body": "\n## Definition\n\nLet $V$ be a [[Hausdorff|0MXB4CKHFW]], locally compact topological space with a countable basis of open sets.\nA *stratification* of $V$ is a locally finite partition $\\mathcal{S}$ of $V$ into locally closed subsets, called strata.\nEach stratum must be a [[smooth manifold|F45VQ47GPV]] in the subspace topology.\nThe frontier condition must hold: if $X, Y \\in \\mathcal{S}$ and $X \\cap \\overline{Y} \\neq \\emptyset$, then $X \\subseteq \\overline{Y}$.\nOne writes $X \u003c Y$ when $X \\ne Y$ and $X \\subseteq \\overline{Y}$.\n*Control data* for $\\mathcal{S}$ consist of three things for each stratum $X$.\nThey are an [[open|02BH3MCBQB]] neighbourhood $T_X$ of $X$ in $V$, a continuous retraction $\\pi_X \\colon T_X \\to X$, and a continuous function $\\rho_X \\colon T_X \\to [0, \\infty)$ with $\\rho_X^{-1}(0) = X$.\nA *Thom–Mather stratified space* is a triple $(V, \\mathcal{S}, \\mathcal{J})$, where $\\mathcal{J}$ is control data for $\\mathcal{S}$, such that:\n\n1. For all strata $X \u003c Y$, the map $(\\pi_X, \\rho_X) \\colon T_X \\cap Y \\to X \\times (0, \\infty)$ is a smooth [[submersion|GTMT5ZFV00]].\n2. For all strata $X \u003c Y$ and every stratum $Z$, the control conditions $\\pi_X(\\pi_Y(v)) = \\pi_X(v)$ and $\\rho_X(\\pi_Y(v)) = \\rho_X(v)$ hold for all $v \\in T_X \\cap T_Y \\cap Z$ for which both sides are defined.\n\n## Remarks\n\nExample: let $V \\subseteq \\R^2$ be the union of the two coordinate axes, with strata $X = \\set{0}$ and the four open half-axes.\nTake $T_X = V$, $\\pi_X(v) = 0$, and $\\rho_X(v) = \\abs{v}^2$.\nOn each half-axis, $\\rho_X$ has non-zero derivative, so $(\\pi_X, \\rho_X)$ is a submersion onto $X \\times (0, \\infty)$.\nFor each half-axis $Y$ take $T_Y = Y$, $\\pi_Y = \\id_Y$ and $\\rho_Y = 0$.\nThen the only control conditions to check are $\\pi_X(\\pi_Y(v)) = 0 = \\pi_X(v)$ and $\\rho_X(\\pi_Y(v)) = \\rho_X(v)$ for $v \\in Y$, which hold since $\\pi_Y(v) = v$.\n\n\nNon-example: give $V = \\R$ the strata $X = \\set{0}$ and $Y = \\R \\setminus \\set{0}$, with $T_X = \\R$ and $\\pi_X = 0$.\nLet $\\rho_X(v) = \\abs{v \\sin(1/v)}$ for $v \\ne 0$ and $\\rho_X(0) = 0$; this function is continuous.\nIt vanishes at the points $v = 1/(k\\pi)$ of $Y$ for every integer $k \\ne 0$, so $\\rho_X^{-1}(0) \\ne X$ and these are not control data.\n\nBy a theorem of Mather, every closed subset of a smooth manifold with a Whitney stratification admits control data that make it a Thom–Mather stratified space.\n",
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