{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is *hyperconnected*, or *irreducible*, if the following two conditions hold.\n\n1. $X$ is nonempty.\n2. For all nonempty [[open sets|02BH3MCBQB]] $U, V \\subseteq X$, the intersection $U \\cap V$ is nonempty.\n\n## Remarks\n\nEquivalent forms: every nonempty open subset of $X$ is [[dense|0BSJSX7ME2]] in $X$.\nAlso equivalent: $X$ is nonempty and is not the union of two closed subsets different from $X$.\n\nExample: an infinite set $X$ with the [[cofinite topology|0B906ZBTMZ]].\nNonempty open sets have finite complements, so the union of the complements of $U$ and $V$ is finite and cannot be all of $X$.\n\nNon-example: a [[Hausdorff space|0S1FJ8VEX9]] with at least two points fails clause 2.\nTwo distinct points have disjoint open neighbourhoods.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "38CDCK1RYB",
    "kind": "definition",
    "msc": [
      "54D05",
      "14A05"
    ],
    "name": "Hyperconnected space",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyperconnected_space"
      }
    ],
    "tag": "5HDNJWZ6K9",
    "uses": [
      "0CQHFPGJ45",
      "02BH3MCBQB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/5HDNJWZ6K9/",
  "links": [
    {
      "field": "defines",
      "target": "38CDCK1RYB"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/5HDNJWZ6K9.md",
  "msc": [
    "54D05",
    "14A05"
  ],
  "name": "Hyperconnected space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5HD/5HDNJWZ6K9.md",
  "status": null,
  "tag": "5HDNJWZ6K9",
  "uri": "m2://5HDNJWZ6K9",
  "url": "https://math2.tamnd.com/wiki/hyperconnected-space-definition/",
  "writing": {
    "tag": "5HDNJWZ6K9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}