{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "6SZDXD3KYH"
    },
    {
      "field": "uses",
      "target": "F55M92H5JY"
    },
    {
      "field": "uses",
      "target": "HH2EM3DPNK"
    },
    {
      "field": "defines",
      "target": "N3Z69FTQ7J"
    }
  ],
  "body": "\n## Summary\n\nA semianalytic set is a subset of a [[real analytic manifold|Z7H98E1NHP]] with a simple local description.\nNear each point it is given by finitely many equations and strict inequalities between [[real analytic functions|081WKEDK6C]].\nIt is the analytic analogue of a [[semialgebraic set|BX5WSK8ZCT]], with polynomials replaced by real analytic functions.\nSemianalytic sets are locally finite unions of connected real analytic submanifolds, and they can be triangulated.\nSince images of semianalytic sets under proper analytic maps need not be semianalytic, one also studies the larger class of [[subanalytic sets|K1HTQF26MA]].\n\n## Explanations\n\n### L1 human\n\nTake some real analytic functions, such as polynomials, $\\sin$ or $\\exp$, and form sets by equations $f = 0$ and inequalities $g \u003e 0$, combined with \"and\", \"or\" and \"not\".\nA semianalytic set is a set that can be described this way near every point.\nFor example, the open half disc $\\set{(x, y) : y \u003e 0,\\ x^2 + y^2 \u003c 1}$ is semianalytic in $\\R^2$.\nThe set $\\set{0} \\cup \\set{1/n : n \\ge 1}$ is not, because near $0$ it consists of infinitely many isolated points.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32B20",
      "14P15"
    ],
    "name": "Semianalytic set",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subanalytic_set"
      }
    ],
    "tag": "E1V22QQHYJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/E1V22QQHYJ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/E1V22QQHYJ.md",
  "msc": [
    "32B20",
    "14P15"
  ],
  "name": "Semianalytic set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/E1V/E1V22QQHYJ.md",
  "status": null,
  "tag": "E1V22QQHYJ",
  "uri": "m2://E1V22QQHYJ",
  "url": "https://tamnd.github.io/math2/wiki/semianalytic-set/",
  "writing": {
    "tag": "E1V22QQHYJ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}