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  "body": "\n## Statement\n\nLet $X \\subseteq \\R^n$ be a [[subanalytic set|K1HTQF26MA]].\nThen the complement $\\R^n \\setminus X$ is subanalytic.\n\n## Remarks\n\nFor [[semianalytic sets|E1V22QQHYJ]] the analogous fact is immediate, since the complement of a set defined by equations and inequalities is again so defined.\nA subanalytic set is locally a projection, and the complement of a projection is not visibly a projection.\nOvercoming this is the content of the theorem; Gabrielov's proof studies the fibres of projections of semianalytic sets.\nFor example, if $A \\subseteq \\R^{n+m}$ is bounded and semianalytic and $\\pi$ forgets the last $m$ coordinates, then $\\R^n \\setminus \\pi(A)$ is subanalytic.\nWith the theorem, the subanalytic subsets of $\\R^n$ form a Boolean algebra, and the closure, interior and boundary of a subanalytic set are subanalytic.\n\n## Explanations\n\n### L1 human\n\nSubanalytic sets are built by taking shadows of sets cut out by analytic equations and inequalities.\nThe theorem says that the region outside such a shadow is again a set of the same kind.\nSo one can take complements freely and never leave the class.\n\n### L2 human\n\nIn semialgebraic geometry, closure under projection is the [[Tarski–Seidenberg theorem|3W5JGDZAN0]], and closure under complement is trivial.\nFor subanalytic sets the roles are reversed: projections are built in, and complements are the hard part.\nGabrielov's theorem is the starting point of the result that the real field with restricted analytic functions is an [[o-minimal|DTWHAW1N0V]] structure.\n",
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