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  "body": "\n## Summary\n\nA Thom–Mather stratified space is a space cut into smooth manifolds, the strata, together with control data near each stratum.\nThe control data are a tubular neighbourhood, a retraction onto the stratum and a distance function to it.\nThe control data must be compatible between strata.\nMather showed that every closed [[Whitney stratified|3F84AMVJNV]] subset of a smooth manifold carries such data.\nThe control data are what makes the [[Thom–Mather isotopy lemmas|7G1PCS2RPG]] work, so these spaces are locally topologically trivial along each stratum.\n\n## Explanations\n\n### L1 human\n\nThink of a cone: it has a tip and a smooth part, two pieces of different dimension.\nNear the tip one can measure the distance to the tip and project nearby points to it.\nA Thom–Mather structure asks for such a projection and such a distance near every piece, and asks the projections and distances of different pieces to agree with each other.\nWith these data one can build vector fields that respect all the pieces, and flow along them.\nThis is how one proves that the space looks the same at all points of a stratum.\n\n",
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