{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "44KMEHYN1R"
    }
  ],
  "body": "\n## Counterexample\n\nThe Sorgenfrey line $\\R_\\ell$ is the set $\\R$ with the topology generated by the base of half-open intervals $[a, b)$ with $a \u003c b$.\nIt is [[separable|0Z2P41SVDP]] and [[first-countable|0CP5NDD21C]], but it is not [[second-countable|03PPNCM9GG]] and not [[metrizable|091GJ3FYD7]].\n\n## Remarks\n\nThe rationals are dense, since each $[a, b)$ contains a rational number.\nThe intervals $[x, x + 1/n)$ for $n \\ge 1$ form a countable neighbourhood base at $x$.\nLet $\\mathcal{B}$ be any base.\nFor each $x \\in \\R$ choose $B_x \\in \\mathcal{B}$ with $x \\in B_x \\subseteq [x, x + 1)$.\nThen $x$ is the least element of $B_x$, so $x \\mapsto B_x$ is injective and $\\mathcal{B}$ is uncountable.\nA separable metrizable space is second-countable, so $\\R_\\ell$ is not metrizable.\nThe Sorgenfrey line is finer than the usual topology, it is [[Lindelöf|SW0SN1NEJE]] and normal, and it is [[totally disconnected|0RJSBR9GW1]] because each $[a, b)$ is both open and closed.\n\n## Explanations\n\n### L1 human\n\nHere an open set around a point may stretch to the right of the point but must start exactly at it.\nThis small change to the real line breaks the link between \"countable dense set\" and \"countable base\".\nIt is the standard example that separable spaces need not be metrizable even when they are very regular.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "counterexample",
    "msc": [
      "54G20",
      "54A10"
    ],
    "name": "Sorgenfrey line",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lower_limit_topology"
      }
    ],
    "tag": "Y4T3GVK8GV",
    "uses": [
      "0Z2P41SVDP",
      "0CP5NDD21C",
      "03PPNCM9GG",
      "091GJ3FYD7",
      "SW0SN1NEJE",
      "0RJSBR9GW1"
    ]
  },
  "kind": "counterexample",
  "link": "https://math2.tamnd.com/node/Y4T3GVK8GV/",
  "links": [
    {
      "field": "uses",
      "target": "0Z2P41SVDP"
    },
    {
      "field": "uses",
      "target": "0CP5NDD21C"
    },
    {
      "field": "uses",
      "target": "03PPNCM9GG"
    },
    {
      "field": "uses",
      "target": "091GJ3FYD7"
    },
    {
      "field": "uses",
      "target": "SW0SN1NEJE"
    },
    {
      "field": "uses",
      "target": "0RJSBR9GW1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y4T3GVK8GV.md",
  "msc": [
    "54G20",
    "54A10"
  ],
  "name": "Sorgenfrey line",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y4T/Y4T3GVK8GV.md",
  "status": null,
  "tag": "Y4T3GVK8GV",
  "uri": "m2://Y4T3GVK8GV",
  "url": "https://math2.tamnd.com/wiki/sorgenfrey-line/",
  "writing": {
    "tag": "Y4T3GVK8GV",
    "kind": "counterexample",
    "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
      },
      {
        "what": "edge refutes or not_instance_of out",
        "have": 0,
        "want": 1
      }
    ]
  }
}