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  "body": "\n## Definition\n\nLet $X$ be a closed subset of $\\R^N$.\nLet $S$ and $S'$ be disjoint smooth [[submanifolds|VKFW27WXEZ]] of $\\R^N$ and let $y \\in S'$.\nThe pair $(S, S')$ satisfies *Whitney's condition (b)* at $y$ if the following holds.\nTake any sequences $x_k \\in S$ and $y_k \\in S'$ that both converge to $y$, with $x_k \\ne y_k$.\nIf the [[tangent spaces|WY6K3P4YCG]] $T_{x_k} S$ converge to a subspace $\\tau$ and the lines through $x_k$ and $y_k$ converge to a line $\\ell$, then $\\ell \\subseteq \\tau$.\nThe pair satisfies *condition (a)* at $y$ if the following holds.\nFor every sequence $x_k \\in S$ converging to $y$ such that $T_{x_k} S$ converges to a subspace $\\tau$, one has $T_y S' \\subseteq \\tau$.\n\nA *Whitney stratification* of $X$ is a locally finite partition of $X$ into connected smooth submanifolds of $\\R^N$, called *strata*, such that:\n\n1. each stratum is locally closed in $X$;\n2. if a stratum $S'$ meets the closure $\\overline{S}$ of a stratum $S \\ne S'$, then $S' \\subseteq \\overline{S} \\setminus S$ (the frontier condition);\n3. every such pair $(S, S')$ satisfies condition (b) at every point of $S'$.\n\n## Remarks\n\nCondition (b) implies condition (a).\nThe same definition applies to subsets of $\\C^N = \\R^{2N}$, and for complex analytic sets one asks the strata to be complex submanifolds.\n\nExample: the cone $X = \\set{x^2 + y^2 = z^2} \\subset \\R^3$ with three strata: $\\set{0}$ and the two connected components of $X \\setminus \\set{0}$, where $z \u003e 0$ and $z \u003c 0$.\nThe frontier condition holds, and condition (b) holds at $0$.\nIndeed, the cone is a union of lines through $0$, so the line from $0$ to $x_k$ lies in the tangent plane $T_{x_k} X$, hence also in the limit plane.\n\nNon-example: the complex Whitney umbrella $X = \\set{x^2 = z y^2} \\subset \\C^3$ with the two strata $S'$, the $z$-axis, and $S = X \\setminus S'$.\nThe surface $S$ is smooth, since the gradient $(2x, -2zy, -y^2)$ of $x^2 - z y^2$ is nonzero off the $z$-axis.\nEvery point $(0, 0, c)$ is a limit of points $(\\sqrt{c}\\, t, t, c)$ of $S$ as $t \\to 0$, so the frontier condition holds.\nFor small real $s \u003e 0$, the point $x_s = (s^{3/2}, s^{1/2}, s^2)$ lies on $S$ and tends to $0$.\nThere the normal vector is $(2 s^{3/2}, -2 s^{5/2}, -s)$, which, divided by $s$, tends to $(0, 0, -1)$.\nSo $T_{x_s} S$ tends to the complex plane $z = 0$, which does not contain $T_0 S'$, the $z$-axis.\nCondition (a) fails at $0$, so condition (b) fails too.\nThe origin must be made a stratum of its own: the strata $S$, the $z$-axis minus $0$, and $\\set{0}$ form a Whitney stratification.\n",
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