{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], and let $\\mathrm{CL}(X)$ be the set of nonempty [[closed|0ZCK8DQYN1]] subsets of $X$.\nFor an [[open|02BH3MCBQB]] set $U \\subseteq X$, put\n$$U^{+} = \\set{A \\in \\mathrm{CL}(X) : A \\subseteq U}, \\qquad U^{-} = \\set{A \\in \\mathrm{CL}(X) : A \\cap U \\ne \\varnothing}.$$\nThe *Vietoris topology* on $\\mathrm{CL}(X)$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $U^{+}$ and $U^{-}$ with $U$ open in $X$.\n\n## Remarks\n\nThe same subbasis, intersected with a smaller family, defines the Vietoris topology on that family, for example on the nonempty [[compact|0R6EWFKRCF]] subsets $\\mathrm{K}(X)$.\nIf $X$ is a metric space, the Vietoris topology on $\\mathrm{K}(X)$ is the topology of the Hausdorff metric.\nIf $X$ is compact Hausdorff, then $\\mathrm{CL}(X)$ with the Vietoris topology is compact Hausdorff.\nIf $X$ is $T_1$, the map $x \\mapsto \\set{x}$ is a [[topological embedding|00ZG9Y9TR4]] of $X$ into $\\mathrm{CL}(X)$.\n\nExample: let $X = [0, 1]$, let $A = [0, 1]$, and let $A_n = \\set{k/n : 0 \\le k \\le n}$.\nEach basic neighbourhood of $A$ has the form $U^+ \\cap V_1^- \\cap \\cdots \\cap V_m^-$ with $A \\subseteq U$ and each $V_i$ open and nonempty.\nFor large $n$, the set $A_n$ meets each $V_i$ and lies in $U = X$, so $A_n \\to A$.\n\nNon-example: the sets $\\set{0}$ do not converge to $[0, 1]$, though they lie in every $U^+$ with $[0, 1] \\subseteq U$.\nThe open set $V = (1/2, 1]$ gives a neighbourhood $V^-$ of $[0, 1]$ that does not contain $\\set{0}$.\n\nThe resulting space is called a hyperspace of $X$, and the topology is also called the finite topology or exponential topology.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "BJVRFFJTV6",
    "kind": "definition",
    "msc": [
      "54B20"
    ],
    "name": "Vietoris topology",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hypertopology"
      }
    ],
    "tag": "2CG46QFBHW",
    "uses": [
      "0CQHFPGJ45",
      "0ZCK8DQYN1",
      "02BH3MCBQB",
      "09S8DV34Y0",
      "0R6EWFKRCF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/2CG46QFBHW/",
  "links": [
    {
      "field": "defines",
      "target": "BJVRFFJTV6"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "09S8DV34Y0"
    },
    {
      "field": "uses",
      "target": "0R6EWFKRCF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/2CG46QFBHW.md",
  "msc": [
    "54B20"
  ],
  "name": "Vietoris topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2CG/2CG46QFBHW.md",
  "status": null,
  "tag": "2CG46QFBHW",
  "uri": "m2://2CG46QFBHW",
  "url": "https://math2.tamnd.com/wiki/vietoris-topology-definition/",
  "writing": {
    "tag": "2CG46QFBHW",
    "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
      }
    ]
  }
}