{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $S$ be a set and let $\\mathcal{F}$ be a [[filter|15PF73BKY6]] on $S$.\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $f \\colon S \\to X$ be a map, and let $x \\in X$.\nThe map $f$ *tends to $x$ along $\\mathcal{F}$*, and $x$ is a *limit of $f$ along $\\mathcal{F}$*, if $f^{-1}(U) \\in \\mathcal{F}$ for every [[neighbourhood|0ZCY0Z80B6]] $U$ of $x$.\n\n## Remarks\n\nEquivalently, the image filter $f(\\mathcal{F}) = \\set{B \\subseteq X : f^{-1}(B) \\in \\mathcal{F}}$ [[converges|70M1VAPFN0]] to $x$.\n\nExample: on $S = \\N$ let $\\mathcal{F}$ be the filter of sets that contain $\\set{n : n \\ge N}$ for some $N$; in Mathlib this is `Filter.atTop`.\nA sequence $f(n) = x_n$ tends to $x$ along $\\mathcal{F}$ exactly when for every neighbourhood $U$ of $x$ there is $N$ with $x_n \\in U$ for all $n \\ge N$.\nThis is the usual [[convergence of a sequence|0W4CN9E2W5]].\n\nNon-example: the sequence $f(n) = (-1)^n$ in $\\R$ has no limit along this filter.\nFor $x \\ne 1$, some neighbourhood of $x$ misses $1$, so its preimage misses all even numbers and contains no tail.\nFor $x = 1$, the preimage of $(0, 2)$ is the set of even numbers, which contains no tail either.\n\nExample: for $a \\in \\R$, the sets $V \\setminus \\set{a}$ with $V$ a neighbourhood of $a$ generate a filter on $\\R \\setminus \\set{a}$, and the limit along it is the usual limit $\\lim_{t \\to a} f(t)$.\n\nIf $X$ is [[Hausdorff|0S1FJ8VEX9]] and $\\emptyset \\notin \\mathcal{F}$, then $f$ has at most one limit along $\\mathcal{F}$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "F6QDHKE6J3",
    "kind": "definition",
    "msc": [
      "54A20",
      "40J05"
    ],
    "name": "Filter limit",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Filter_(mathematics)"
      }
    ],
    "tag": "3JGEQYCF6S",
    "uses": [
      "15PF73BKY6",
      "0CQHFPGJ45",
      "0ZCY0Z80B6",
      "70M1VAPFN0",
      "0W4CN9E2W5",
      "0S1FJ8VEX9"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/3JGEQYCF6S/",
  "links": [
    {
      "field": "defines",
      "target": "F6QDHKE6J3"
    },
    {
      "field": "uses",
      "target": "15PF73BKY6"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0ZCY0Z80B6"
    },
    {
      "field": "uses",
      "target": "70M1VAPFN0"
    },
    {
      "field": "uses",
      "target": "0W4CN9E2W5"
    },
    {
      "field": "uses",
      "target": "0S1FJ8VEX9"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3JGEQYCF6S.md",
  "msc": [
    "54A20",
    "40J05"
  ],
  "name": "Filter limit",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3JG/3JGEQYCF6S.md",
  "status": null,
  "tag": "3JGEQYCF6S",
  "uri": "m2://3JGEQYCF6S",
  "url": "https://math2.tamnd.com/wiki/filter-limit-definition/",
  "writing": {
    "tag": "3JGEQYCF6S",
    "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
      }
    ]
  }
}