{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nFor $n \\ge 1$ let $C_n \\subseteq \\R^2$ be the [[circle|060W5CBZP6]] with centre $(1/n, 0)$ and radius $1/n$.\nThe Hawaiian earring is $H = \\bigcup_{n \\ge 1} C_n$ with the subspace topology from $\\R^2$.\nAll circles pass through the origin $0$.\nThe space $H$ is [[compact|0TVJZYB6SH]] and [[path-connected|0WB6DH499T]], but it is not [[semilocally simply connected|R3SHZX0NZR]] at $0$.\n\nCheck.\n$H$ is closed and bounded in $\\R^2$, so it is compact.\nEach $C_n$ is path-connected and contains $0$, so $H$ is path-connected.\nFix $n$ and let $r_n \\colon H \\to C_n$ be the identity on $C_n$ and send every other circle to $0$.\nThis map is continuous, and it is a retraction onto $C_n$.\nLet $\\gamma_n$ be the loop that goes once around $C_n$.\nThe map $r_n$ sends it to a generator of the [[fundamental group|S1XK3HXHFX]] $\\pi_1(C_n) \\cong \\Z$, so $\\gamma_n$ is not null-homotopic in $H$.\nEvery neighbourhood of $0$ contains $C_n$ for all large $n$, so it contains loops at $0$ that are not null-homotopic in $H$.\n\n## Remarks\n\nSince $H$ is not semilocally simply connected, it has no universal [[covering space|3XZRRE165N]].\nThe retractions $r_n$ together give a homomorphism from $\\pi_1(H, 0)$ onto the product $\\prod_{n \\ge 1} \\Z$.\nIt is onto: a loop can run $k_n$ times around $C_n$ during the time interval $[1 - 2^{1-n}, 1 - 2^{-n}]$.\nThis loop is continuous at time $1$ because the circles shrink.\nSo $\\pi_1(H, 0)$ is uncountable.\nThe [[wedge sum|5JSWM1HBX2]] of countably many circles, with the CW topology, has a countable free fundamental group and is not homeomorphic to $H$.\n\n## Explanations\n\n### L1 human\n\nThe Hawaiian earring is an infinite family of circles in the plane, all touching at one point and getting smaller and smaller.\nIt looks like a bouquet of circles, but the small circles crowd near the common point.\nHowever close you look at that point, you still see whole circles there that cannot be shrunk away.\nThis makes its loops far more complicated than those of a finite bouquet.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "example",
    "msc": [
      "55Q52",
      "54G20"
    ],
    "name": "Hawaiian earring",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hawaiian_earring"
      }
    ],
    "tag": "73EBKQPP8Q",
    "uses": [
      "060W5CBZP6",
      "0TVJZYB6SH",
      "0WB6DH499T"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/73EBKQPP8Q/",
  "links": [
    {
      "field": "uses",
      "target": "060W5CBZP6"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "0WB6DH499T"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/73EBKQPP8Q.md",
  "msc": [
    "55Q52",
    "54G20"
  ],
  "name": "Hawaiian earring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/73E/73EBKQPP8Q.md",
  "status": null,
  "tag": "73EBKQPP8Q",
  "uri": "m2://73EBKQPP8Q",
  "url": "https://math2.tamnd.com/wiki/hawaiian-earring/",
  "writing": {
    "tag": "73EBKQPP8Q",
    "kind": "example",
    "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
      },
      {
        "what": "edge instance_of or not_instance_of or illustrates out",
        "have": 0,
        "want": 1
      }
    ]
  }
}