{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]], let $S \\subseteq X$ and let $x \\in X$.\nThe point $x$ is a limit point of $S$ when each [[open set|01ZQQDCERP]] $U$ with $x \\in U$ contains a point of $S$ other than $x$.\n\n## Remarks\n\nThe point $x$ need not be in $S$. In $\\mathbb{R}$, the point $0$ is a limit point of $\\{1/n : n \\ge 1\\}$.\nIn a metric space, $x$ is a limit point of $S$ if and only if each ball $B(x, \\varepsilon)$ contains a point of $S \\setminus \\{x\\}$.\nThe [[closure|0DRKT1QJR7]] of $S$ is the union of $S$ and the set of its limit points.\nA point of $S$ that is not a limit point of $S$ is an [[isolated point|01PKQYTTMT]] of $S$.\n\n## Explanations\n\n### L1 human\n\nA limit point is a point that other points of the set crowd around.\nFor the set $\\{1, 1/2, 1/3, \\ldots\\}$, the points crowd around $0$, so $0$ is a limit point, but $1$ is not.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "defines": "0E3X7TXS67",
    "kind": "definition",
    "msc": [
      "54A05"
    ],
    "name": "Limit point",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Accumulation_point"
      }
    ],
    "tag": "0HBQC2GJXT",
    "uses": [
      "0CNQXVBHCX",
      "01ZQQDCERP"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0HBQC2GJXT/",
  "links": [
    {
      "field": "defines",
      "target": "0E3X7TXS67"
    },
    {
      "field": "uses",
      "target": "0CNQXVBHCX"
    },
    {
      "field": "uses",
      "target": "01ZQQDCERP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HBQC2GJXT.md",
  "msc": [
    "54A05"
  ],
  "name": "Limit point",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HB/0HBQC2GJXT.md",
  "status": null,
  "tag": "0HBQC2GJXT",
  "uri": "m2://0HBQC2GJXT",
  "url": "https://math2.tamnd.com/wiki/limit-point-definition/",
  "writing": {
    "tag": "0HBQC2GJXT",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}