{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "AMVHH8JT14"
    }
  ],
  "body": "\n## Summary\n\nA σ-compact space is a topological space that is a countable union of compact subspaces.\nEuclidean space $\\R^n$ is the main example, as the union of the closed balls of radius $n$.\nEvery σ-compact space is [[Lindelöf|SW0SN1NEJE]], and every [[locally compact|0FE2SB8WVJ]] second-countable space is σ-compact.\nThe notion lets compactness arguments run one piece at a time.\n\n## Explanations\n\n### L1 human\n\nA compact space is small in a topological sense.\nA σ-compact space may be big, but it can be built from countably many compact pieces.\nThe real line is not compact, but it is the union of the intervals $[-n, n]$ for $n = 1, 2, 3, \\ldots$.\nSo many facts proved on compact intervals can be extended to all of $\\R$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "54D45"
    ],
    "name": "Sigma-compact space",
    "same_as": {
      "wikidata": "Q290618",
      "wikipedia": "Σ-compact space"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Σ-compact_space"
      }
    ],
    "specializes": [
      "0CQHFPGJ45"
    ],
    "tag": "ZW4NFH5RNC"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/ZW4NFH5RNC/",
  "links": [
    {
      "field": "specializes",
      "target": "0CQHFPGJ45"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZW4NFH5RNC.md",
  "msc": [
    "54D45"
  ],
  "name": "Sigma-compact space",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZW4/ZW4NFH5RNC.md",
  "status": null,
  "tag": "ZW4NFH5RNC",
  "uri": "m2://ZW4NFH5RNC",
  "url": "https://math2.tamnd.com/wiki/sigma-compact-space/",
  "writing": {
    "tag": "ZW4NFH5RNC",
    "kind": "concept",
    "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
      }
    ]
  }
}