{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01MPZXYVYF"
    },
    {
      "field": "uses",
      "target": "03YEM7JEDN"
    },
    {
      "field": "uses",
      "target": "048F2P90WR"
    },
    {
      "field": "uses",
      "target": "0SSKBQN58M"
    },
    {
      "field": "uses",
      "target": "0YBVKMYX0W"
    }
  ],
  "body": "\n## Definition\n\nLet $D$ be a subset of the [[complex plane|0CTE7A31XQ]] $\\mathbb{C}$.\nThe set $D$ is open if for each $z_0 \\in D$ there is a real number $r \u003e 0$ such that each $z \\in \\mathbb{C}$ with $|z - z_0| \u003c r$ is in $D$. Here $|z - z_0|$ is the [[modulus|0KW2X0TFBX]] of $z - z_0$.\nThe set $D$ is connected if there are no two disjoint nonempty open sets $A$ and $B$ with $D = A \\cup B$.\nA domain in the complex plane is a subset $D \\subseteq \\mathbb{C}$ that is nonempty, open and connected.\n\n## Remarks\n\nAn open set $D \\subseteq \\mathbb{C}$ is connected if and only if each two points of $D$ are the end points of a polygonal path in $D$.\nExamples of domains are $\\mathbb{C}$, each open disc, the upper half-plane $\\{z : \\operatorname{Im} z \u003e 0\\}$ and $\\mathbb{C} \\setminus \\{0\\}$.\nThe union of two disjoint open discs is open but not connected, so it is not a domain.\nSome authors use the word region for a domain. Other authors use region for a set between a domain and its closure.\n\n## Explanations\n\n### L1 human\n\nA domain is a set in the plane that has no boundary points in it and that is in one piece.\nOpen means that you can move a little in each direction from each point and stay in the set.\nConnected means that you can walk from each point of the set to each other point without leaving the set.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0EWMDD6C14",
    "kind": "definition",
    "msc": [
      "30A99"
    ],
    "name": "Domain in the complex plane",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Domain_(mathematical_analysis)"
      },
      {
        "locator": "Conventions",
        "url": "https://en.wikipedia.org/wiki/Domain_(mathematical_analysis)"
      }
    ],
    "tag": "097KNJFJ7V",
    "uses": [
      "0CTE7A31XQ",
      "0KW2X0TFBX"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/097KNJFJ7V/",
  "links": [
    {
      "field": "defines",
      "target": "0EWMDD6C14"
    },
    {
      "field": "uses",
      "target": "0CTE7A31XQ"
    },
    {
      "field": "uses",
      "target": "0KW2X0TFBX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/097KNJFJ7V.md",
  "msc": [
    "30A99"
  ],
  "name": "Domain in the complex plane",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/097/097KNJFJ7V.md",
  "status": null,
  "tag": "097KNJFJ7V",
  "uri": "m2://097KNJFJ7V",
  "url": "https://math2.tamnd.com/wiki/domain-in-the-complex-plane-definition/",
  "writing": {
    "tag": "097KNJFJ7V",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}