{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe separate two points by balls of half their distance.\n\n1. Let $x, y \\in X$ with $x \\neq y$. By the axioms of a [[metric|063W5JMY85]], $d(x, y) \u003e 0$. Let $r = d(x, y) / 2 \u003e 0$.\n2. The [[open balls|0DC3MPR7C9]] $B(x, r)$ and $B(y, r)$ are [[open|0B2305GNAZ]]. They contain $x$ and $y$, since $d(x, x) = 0 \u003c r$ and $d(y, y) = 0 \u003c r$.\n3. Suppose some $z$ lies in both balls. Then $d(x, z) \u003c r$ and $d(z, y) \u003c r$.\n4. By the [[triangle inequality|01M6CW9VM6]], $d(x, y) \\le d(x, z) + d(z, y) \u003c 2r = d(x, y)$. This is impossible.\n5. So $B(x, r) \\cap B(y, r) = \\varnothing$, and $X$ is [[Hausdorff|0MXB4CKHFW]].\n\n## Idea\n\nTwo distinct points are at a positive distance. Balls of half that radius around them cannot meet, since a common point would give a path between them that is shorter than their distance.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "54E35"
    ],
    "name": "Proof that metric spaces are Hausdorff",
    "proves": "0P9DCDES73",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Metric_Space_is_Hausdorff"
      }
    ],
    "tag": "NK3WS9ENHG",
    "uses": [
      "0MXB4CKHFW",
      "0DC3MPR7C9",
      "0B2305GNAZ",
      "01M6CW9VM6",
      "063W5JMY85"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/NK3WS9ENHG/",
  "links": [
    {
      "field": "proves",
      "target": "0P9DCDES73"
    },
    {
      "field": "uses",
      "target": "0MXB4CKHFW"
    },
    {
      "field": "uses",
      "target": "0DC3MPR7C9"
    },
    {
      "field": "uses",
      "target": "0B2305GNAZ"
    },
    {
      "field": "uses",
      "target": "01M6CW9VM6"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NK3WS9ENHG.md",
  "msc": [
    "54E35"
  ],
  "name": "Proof that metric spaces are Hausdorff",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NK3/NK3WS9ENHG.md",
  "status": null,
  "tag": "NK3WS9ENHG",
  "uri": "m2://NK3WS9ENHG",
  "url": "https://math2.tamnd.com/wiki/proof-that-metric-spaces-are-hausdorff/",
  "writing": {
    "tag": "NK3WS9ENHG",
    "kind": "proof",
    "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
      }
    ]
  }
}