{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "D1T39KHPDN"
    },
    {
      "field": "uses",
      "target": "D1T39KHPDN"
    }
  ],
  "body": "\n## Definition\n\nLet $L$ be a set with more than one element and let $\u003c$ be a [[total order|0FRXBVVR03]] on $L$.\nThe ordered set $(L, \u003c)$ is a *linear continuum* if:\n\n1. Every nonempty subset of $L$ that is bounded above has a least upper bound, a [[supremum|0R3H5PANRQ]], in $L$.\n2. For all $x, y \\in L$ with $x \u003c y$, there is $z \\in L$ with $x \u003c z \u003c y$.\n\n## Remarks\n\nThe real line $\\R$ is a linear continuum: clause 1 is the completeness of $\\R$, and for clause 2 one takes $z = (x + y)/2$.\nThe integers $\\Z$ satisfy clause 1, but they fail clause 2, since there is no integer strictly between $1$ and $2$.\nThe rationals $\\Q$ satisfy clause 2 but fail clause 1: the set $\\set{q \\in \\Q : q^2 \u003c 2}$ is bounded above but has no least upper bound in $\\Q$.\nThe square $[0, 1] \\times [0, 1]$ with the dictionary order is a linear continuum that is not order isomorphic to an interval of $\\R$.\nA linear continuum and each of its intervals are [[connected|0V032VWTHD]] in the order topology.\n\n## Explanations\n\n### L1 human\n\nA linear continuum is an ordered set with no gaps and no jumps.\n\"No gaps\" means that bounded sets have a least upper bound, and \"no jumps\" means that between two elements there is always a third one.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "MM0QPT0GT0",
    "kind": "definition",
    "msc": [
      "54F05",
      "06A05"
    ],
    "name": "Linear continuum",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Linear_continuum"
      }
    ],
    "tag": "MCDP25YMSE",
    "uses": [
      "0FRXBVVR03",
      "0R3H5PANRQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/MCDP25YMSE/",
  "links": [
    {
      "field": "defines",
      "target": "MM0QPT0GT0"
    },
    {
      "field": "uses",
      "target": "0FRXBVVR03"
    },
    {
      "field": "uses",
      "target": "0R3H5PANRQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MCDP25YMSE.md",
  "msc": [
    "54F05",
    "06A05"
  ],
  "name": "Linear continuum",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MCD/MCDP25YMSE.md",
  "status": null,
  "tag": "MCDP25YMSE",
  "uri": "m2://MCDP25YMSE",
  "url": "https://math2.tamnd.com/wiki/linear-continuum-definition/",
  "writing": {
    "tag": "MCDP25YMSE",
    "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
      }
    ]
  }
}