{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[Hausdorff space|0MXB4CKHFW]] and let $(x_n)_{n \\ge 1}$ be a sequence in $X$.\nSay that $(x_n)$ converges to $x \\in X$ when for each [[neighbourhood|09CJ00H3NW]] $N$ of $x$ there is $n_0$ with $x_n \\in N$ for all $n \\ge n_0$.\nIf $(x_n)$ converges to $x$ and to $y$, then $x = y$.\n\n## Remarks\n\nProof: if $x \\neq y$, take disjoint open sets $U \\ni x$ and $V \\ni y$. For large $n$, the point $x_n$ is in both $U$ and $V$, which is impossible.\nThe same holds for nets and filters. In fact, a space is Hausdorff if and only if each net in it has at most one limit.\nIn an [[indiscrete space|0XJ02JBD6S]] with at least two points, each sequence converges to each point.\n\n## Explanations\n\n### L1 human\n\nA sequence cannot come close to two different points at the same time, if the points have disjoint neighbourhoods.\nSo in Hausdorff spaces, you can speak of the limit of a sequence.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "kind": "statement",
    "msc": [
      "54D10"
    ],
    "name": "Limits are unique in Hausdorff spaces",
    "role": "proposition",
    "sources": [
      {
        "locator": "Equivalences",
        "url": "https://en.wikipedia.org/wiki/Hausdorff_space"
      }
    ],
    "tag": "0D8T984XEE",
    "uses": [
      "0MXB4CKHFW",
      "09CJ00H3NW"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0D8T984XEE/",
  "links": [
    {
      "field": "uses",
      "target": "0MXB4CKHFW"
    },
    {
      "field": "uses",
      "target": "09CJ00H3NW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0D8T984XEE.md",
  "msc": [
    "54D10"
  ],
  "name": "Limits are unique in Hausdorff spaces",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0D8/0D8T984XEE.md",
  "status": null,
  "tag": "0D8T984XEE",
  "uri": "m2://0D8T984XEE",
  "url": "https://math2.tamnd.com/wiki/limits-are-unique-in-hausdorff-spaces/",
  "writing": {
    "tag": "0D8T984XEE",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}