{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a set.\nA pseudometric on $X$ is a function $d \\colon X \\times X \\to \\mathbb{R}$ such that, for all $x, y, z \\in X$:\n\n1. $d(x, x) = 0$.\n2. $d(x, y) = d(y, x)$.\n3. $d(x, z) \\le d(x, y) + d(y, z)$.\n\nA pseudometric space is a pair $(X, d)$, where $X$ is a set and $d$ is a pseudometric on $X$.\n\n## Remarks\n\nEach [[metric|063W5JMY85]] is a pseudometric. A pseudometric is a metric if and only if $d(x, y) = 0$ implies $x = y$.\nOn the set of functions $f \\colon [0, 1] \\to \\mathbb{R}$, the rule $d(f, g) = |f(0) - g(0)|$ is a pseudometric, but not a metric.\nThe relation $d(x, y) = 0$ is an equivalence relation, and $d$ gives a metric on the set of its classes.\n\n## Explanations\n\n### L1 human\n\nA pseudometric measures distance, but it can fail to see the difference between two points.\nFor example, measure the distance between two people by the difference of their ages. Then two people of the same age have distance zero.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0A3HJX3KTF",
    "kind": "definition",
    "msc": [
      "54E35"
    ],
    "name": "Pseudometric space",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Pseudometric_space"
      }
    ],
    "tag": "0H3W614GEC",
    "uses": [
      "063W5JMY85"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0H3W614GEC/",
  "links": [
    {
      "field": "defines",
      "target": "0A3HJX3KTF"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0H3W614GEC.md",
  "msc": [
    "54E35"
  ],
  "name": "Pseudometric space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0H3/0H3W614GEC.md",
  "status": null,
  "tag": "0H3W614GEC",
  "uri": "m2://0H3W614GEC",
  "url": "https://math2.tamnd.com/wiki/pseudometric-space-definition/",
  "writing": {
    "tag": "0H3W614GEC",
    "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
      }
    ]
  }
}