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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $X \\subseteq \\R^n$.\nThe set $X$ is *subanalytic* if each point $a \\in \\R^n$ has an open neighbourhood $U$ with the following property.\nThere are an integer $m \\ge 0$ and a bounded [[semianalytic set|E1V22QQHYJ]] $A \\subseteq \\R^{n+m}$ such that $X \\cap U = \\pi(A)$.\nHere $\\pi \\colon \\R^{n+m} \\to \\R^n$ is the [[projection|033DA1WE8Y]] onto the first $n$ coordinates.\n\n## Remarks\n\nEvery semianalytic set $X \\subseteq \\R^n$ is subanalytic.\nIndeed, for a point $a$ take an open ball $U$ around $a$, take $m = 0$ and $A = X \\cap U$.\nThis set $A$ is semianalytic and bounded, and $\\pi$ is the identity, so $X \\cap U = \\pi(A)$.\n\nThe set $X = \\set{1/k : k = 1, 2, 3, \\dots} \\subseteq \\R$ is not subanalytic.\nA subanalytic set meets each small enough neighbourhood of a point in finitely many connected components.\nBut every neighbourhood of $0$ contains infinitely many points of $X$, each of them a connected component of the intersection.\n\nThe same definition works in a real analytic manifold $M$.\nOne replaces $\\R^m$ by another real analytic manifold $N$ and asks for a relatively compact semianalytic subset $A$ of $M \\times N$.\n",
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