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  "body": "\n## Statement\n\nLet $X \\subseteq \\R^n$ be a closed [[subanalytic set|K1HTQF26MA]] and let $A_1, \\dots, A_k$ be subanalytic subsets of $X$.\nThen there are a locally finite [[simplicial complex|V8GZ0W1ZZ5]] $K$ and a [[homeomorphism|0QT00PB5G2]] $h \\colon \\abs{K} \\to X$ from the polyhedron of $K$ onto $X$ with the following property.\nEach $A_j$ is a union of sets $h(\\sigma^\\circ)$, where $\\sigma^\\circ$ is the interior of a simplex $\\sigma$ of $K$.\n\n## Remarks\n\nŁojasiewicz proved the statement for [[semianalytic sets|E1V22QQHYJ]], and Hironaka extended it to subanalytic sets.\nThe proof first decomposes $X$ into finitely many pieces on which suitable coordinate projections are finite and well behaved, and then triangulates by induction on $n$.\nEvery closed real analytic subset of $\\R^n$ is semianalytic, and every [[complex analytic set|GB7M582M5Y]] in $\\C^n = \\R^{2n}$ is a closed real analytic subset.\nSo both kinds of analytic sets are [[triangulable|YH170WBQ6M]], and a triangulation can be chosen compatible with the singular locus.\nFor instance, the cone $\\set{x^2 + y^2 = z^2}$ in $\\R^3$ can be triangulated with the vertex at the origin as a vertex of $K$.\n\n## Explanations\n\n### L1 human\n\nAn analytic set can have singular points, such as the tip of a cone or a crossing of two curves.\nThe theorem says that, however complicated it is, the set can be cut into pieces that are curved copies of points, segments, triangles and their higher analogues, glued along faces.\nThis makes analytic sets accessible to combinatorial topology.\n\n### L2 human\n\nTriangulability implies that a compact analytic set has finitely generated homology and a well defined Euler characteristic.\nIt also implies local conic structure: a small neighbourhood of a point of an analytic set is homeomorphic to the cone over its link.\nThe choice of a triangulation compatible with the subsets $A_j$ lets one treat a set together with a stratification of it.\n",
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