{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $Y$ be a topological space and $A \\subseteq Y$.\nA *retraction* of $Y$ onto $A$ is a [[continuous|0VGFHTM75Y]] map $r \\colon Y \\to A$ with $r(a) = a$ for all $a \\in A$.\n\nLet $X$ be a [[metrizable space|091GJ3FYD7]].\n$X$ is an *absolute neighborhood retract* (ANR) if the following holds for every metrizable space $Y$.\nLet $h$ be any [[homeomorphism|0QT00PB5G2]] of $X$ onto a [[closed|0ZCK8DQYN1]] subset $h(X)$ of $Y$.\nThen there is an [[open|02BH3MCBQB]] set $U \\subseteq Y$ with $h(X) \\subseteq U$ and a retraction of $U$ onto $h(X)$.\n$X$ is an *absolute retract* (AR) if one can always take $U = Y$.\n\n## Remarks\n\nSome authors define these classes for compact metrizable spaces only, or for other classes of spaces; the definition above is for metrizable spaces.\nEquivalently, $X$ is an ANR if every continuous map $f \\colon A \\to X$ from a closed subset $A$ of a metrizable space $Y$ extends to a continuous map on some open set $U \\supseteq A$.\n\nExample: $X = [0, 1]$ is an AR.\nGiven $h$ as above, the map $h^{-1} \\colon h(X) \\to [0, 1]$ is continuous on the closed set $h(X)$, and by the [[Tietze extension theorem|0X52TNMTCJ]] it extends to a continuous $g \\colon Y \\to [0, 1]$.\nThen $r = h \\circ g$ is a retraction of $Y$ onto $h(X)$.\n\nNon-example: $X = \\set{0} \\cup \\set{1/n : n \\ge 1}$, a closed subset of $Y = \\R$, with $h$ the inclusion.\nAny open $U \\supseteq X$ contains an interval $(-\\varepsilon, \\varepsilon)$.\nA retraction $r \\colon U \\to X$ would map this connected interval onto a connected subset of $X$ that contains $0$ and $1/n$ for large $n$, which is impossible.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "DK3MT3HYX5",
    "kind": "definition",
    "msc": [
      "54C55",
      "54C15"
    ],
    "name": "Absolute neighborhood retract",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Absolute_neighborhood_retract"
      }
    ],
    "tag": "72SN9MMH5Y",
    "uses": [
      "091GJ3FYD7",
      "0QT00PB5G2",
      "0ZCK8DQYN1",
      "02BH3MCBQB",
      "0VGFHTM75Y",
      "0X52TNMTCJ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/72SN9MMH5Y/",
  "links": [
    {
      "field": "defines",
      "target": "DK3MT3HYX5"
    },
    {
      "field": "uses",
      "target": "091GJ3FYD7"
    },
    {
      "field": "uses",
      "target": "0QT00PB5G2"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "0VGFHTM75Y"
    },
    {
      "field": "uses",
      "target": "0X52TNMTCJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/72SN9MMH5Y.md",
  "msc": [
    "54C55",
    "54C15"
  ],
  "name": "Absolute neighborhood retract",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/72S/72SN9MMH5Y.md",
  "status": null,
  "tag": "72SN9MMH5Y",
  "uri": "m2://72SN9MMH5Y",
  "url": "https://math2.tamnd.com/wiki/absolute-neighborhood-retract-definition/",
  "writing": {
    "tag": "72SN9MMH5Y",
    "kind": "definition",
    "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
      }
    ]
  }
}