{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00R0X1WGPX"
    },
    {
      "field": "uses",
      "target": "065J822CKN"
    },
    {
      "field": "uses",
      "target": "08SVRVHJNY"
    },
    {
      "field": "uses",
      "target": "0K3T4A659R"
    },
    {
      "field": "uses",
      "target": "0P6PMJB4MA"
    },
    {
      "field": "uses",
      "target": "0R37DFJKFW"
    }
  ],
  "body": "\n## Definition\n\nLet $(X, \\tau)$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\nThe subspace topology on $S$ is\n$$\\tau_S = \\{ S \\cap U : U \\in \\tau \\}.$$\nWith this topology, $S$ is a subspace of $X$.\n\n## Remarks\n\nThe set $\\tau_S$ is a topology on $S$, because intersection with $S$ commutes with unions and finite [[intersections|0HBRW4E6G1]].\nA set open in $S$ need not be open in $X$. For example, $[0, 1)$ is open in the subspace $[0, 1]$ of $\\mathbb{R}$, because $[0, 1) = [0, 1] \\cap (-1, 1)$.\nFor a metric space, the subspace topology is the metric topology of the [[subspace metric|0T2WYDTRP2]].\nThe subspace topology is the coarsest topology on $S$ for which the [[inclusion map is continuous|065J822CKN]].\n\n## Explanations\n\n### L1 human\n\nA subset of a space inherits a topology: you cut each open set of the big space down to the subset.\nThis is how the circle and the interval get their topologies from the plane and the line.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0WS9P12FQR",
    "kind": "definition",
    "msc": [
      "54B05"
    ],
    "name": "Subspace topology",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Subspace_topology"
      }
    ],
    "tag": "0D5YHQGDJY",
    "uses": [
      "0CNQXVBHCX",
      "0HBRW4E6G1",
      "0FE8Q5S42Y"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0D5YHQGDJY/",
  "links": [
    {
      "field": "defines",
      "target": "0WS9P12FQR"
    },
    {
      "field": "uses",
      "target": "0CNQXVBHCX"
    },
    {
      "field": "uses",
      "target": "0HBRW4E6G1"
    },
    {
      "field": "uses",
      "target": "0FE8Q5S42Y"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0D5YHQGDJY.md",
  "msc": [
    "54B05"
  ],
  "name": "Subspace topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0D5/0D5YHQGDJY.md",
  "status": null,
  "tag": "0D5YHQGDJY",
  "uri": "m2://0D5YHQGDJY",
  "url": "https://math2.tamnd.com/wiki/subspace-topology-definition/",
  "writing": {
    "tag": "0D5YHQGDJY",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}