{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0K3T4A659R"
    },
    {
      "field": "uses",
      "target": "0MXN7AFBPE"
    },
    {
      "field": "uses",
      "target": "0SB9FNZECG"
    },
    {
      "field": "uses",
      "target": "0ZMT1GP4KM"
    }
  ],
  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nA path in $X$ from $x$ to $y$ is a [[continuous map|0SY9ECQ19Q]] $\\gamma \\colon [0, 1] \\to X$ with $\\gamma(0) = x$ and $\\gamma(1) = y$.\nThe space $X$ is path-connected when $X$ is nonempty and for all $x, y \\in X$ there is a path in $X$ from $x$ to $y$.\nA subset of $X$ is path-connected when it is path-connected in the [[subspace topology|0D5YHQGDJY]].\n\n## Remarks\n\nSome authors do not require $X$ to be nonempty.\nEach convex subset of $\\mathbb{R}^n$ is path-connected: use the segment $\\gamma(t) = (1 - t)x + ty$.\n[[Path-connected spaces are connected|0ZMT1GP4KM]]. The [[topologist's sine curve|0K3T4A659R]] is connected but not path-connected.\n\n## Explanations\n\n### L1 human\n\nA space is path-connected when you can walk from each point to each other point along a continuous path inside the space.\nThis is a concrete way to show that a space is in one piece.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0WB6DH499T",
    "kind": "definition",
    "msc": [
      "54D05"
    ],
    "name": "Path-connected space",
    "sources": [
      {
        "locator": "Path connectedness",
        "url": "https://en.wikipedia.org/wiki/Connected_space"
      }
    ],
    "tag": "0K0GE6BCYW",
    "uses": [
      "0CNQXVBHCX",
      "0SY9ECQ19Q"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0K0GE6BCYW/",
  "links": [
    {
      "field": "defines",
      "target": "0WB6DH499T"
    },
    {
      "field": "uses",
      "target": "0CNQXVBHCX"
    },
    {
      "field": "uses",
      "target": "0SY9ECQ19Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0K0GE6BCYW.md",
  "msc": [
    "54D05"
  ],
  "name": "Path-connected space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0K0/0K0GE6BCYW.md",
  "status": null,
  "tag": "0K0GE6BCYW",
  "uri": "m2://0K0GE6BCYW",
  "url": "https://math2.tamnd.com/wiki/path-connected-space-definition/",
  "writing": {
    "tag": "0K0GE6BCYW",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}