{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "098C0KZZAP"
    },
    {
      "field": "uses",
      "target": "0ZMT1GP4KM"
    }
  ],
  "body": "\n## Statement\n\nLet $S \\subseteq \\mathbb{R}$, where $\\mathbb{R}$ has the usual topology.\nThen $S$ is [[connected|0R37DFJKFW]] if and only if $S$ is an [[interval|0TK8Y3J5RW]], that is, for all $a, b \\in S$ and all $c \\in \\mathbb{R}$ with $a \u003c c \u003c b$, $c \\in S$.\n\n## Remarks\n\nIf $S$ is not an interval, take $a \u003c c \u003c b$ with $a, b \\in S$ and $c \\notin S$. Then $(-\\infty, c)$ and $(c, \\infty)$ split $S$.\nThe proof that an interval $I$ is connected uses the [[completeness axiom|0AXWET80YW]]. If open sets $U$ and $V$ split $I$, with $a \\in U$, $b \\in V$ and $a \u003c b$, let $c$ be the [[supremum|0BTMQ04556]] of the points of $U \\cap [a, b]$. Then $c$ lies in neither $U$ nor $V$.\nUnder this definition the empty set and single points are intervals, and they are connected.\nThe statement fails for $\\mathbb{Q}$: no interval of $\\mathbb{Q}$ with more than one point is connected.\n\n## Explanations\n\n### L1 human\n\nOn the real line, the sets in one piece are exactly the intervals.\nThe proof needs the completeness of $\\mathbb{R}$, because $\\mathbb{Q}$ has gaps where it can be split.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "54D05"
    ],
    "name": "Connected subsets of the real line are intervals",
    "role": "theorem",
    "sources": [
      {
        "locator": "Examples",
        "url": "https://en.wikipedia.org/wiki/Connected_space"
      }
    ],
    "tag": "0B9ZQFMQNV",
    "uses": [
      "0R37DFJKFW",
      "0TK8Y3J5RW",
      "0BTMQ04556",
      "0AXWET80YW"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0B9ZQFMQNV/",
  "links": [
    {
      "field": "uses",
      "target": "0R37DFJKFW"
    },
    {
      "field": "uses",
      "target": "0TK8Y3J5RW"
    },
    {
      "field": "uses",
      "target": "0BTMQ04556"
    },
    {
      "field": "uses",
      "target": "0AXWET80YW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0B9ZQFMQNV.md",
  "msc": [
    "54D05"
  ],
  "name": "Connected subsets of the real line are intervals",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0B9/0B9ZQFMQNV.md",
  "status": null,
  "tag": "0B9ZQFMQNV",
  "uri": "m2://0B9ZQFMQNV",
  "url": "https://math2.tamnd.com/wiki/connected-subsets-of-the-real-line-are-intervals/",
  "writing": {
    "tag": "0B9ZQFMQNV",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}