{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "11PJJQRECB"
    },
    {
      "field": "uses",
      "target": "5KERJ85BXJ"
    },
    {
      "field": "uses",
      "target": "6P1E2FJPSD"
    },
    {
      "field": "uses",
      "target": "6PWN57BW45"
    },
    {
      "field": "uses",
      "target": "7DM2KSS2DB"
    },
    {
      "field": "uses",
      "target": "9EA7KJ50AE"
    },
    {
      "field": "uses",
      "target": "AXT498MW87"
    },
    {
      "field": "uses",
      "target": "C1T9VYKNV1"
    },
    {
      "field": "uses",
      "target": "DC4JPVPXEJ"
    },
    {
      "field": "uses",
      "target": "F01D990B61"
    },
    {
      "field": "uses",
      "target": "FFH90QRFFJ"
    },
    {
      "field": "uses",
      "target": "G003WR6SMX"
    },
    {
      "field": "uses",
      "target": "GPDPYHM3RE"
    },
    {
      "field": "uses",
      "target": "PDFK69ZMST"
    },
    {
      "field": "uses",
      "target": "QF764XKBA3"
    },
    {
      "field": "uses",
      "target": "R6DXPZPVV7"
    },
    {
      "field": "uses",
      "target": "R7G4ZC0963"
    },
    {
      "field": "uses",
      "target": "SYP3FVM71M"
    },
    {
      "field": "uses",
      "target": "VEGBD6Y61A"
    },
    {
      "field": "uses",
      "target": "WP12H9FG91"
    },
    {
      "field": "uses",
      "target": "XHHTS6TDYZ"
    },
    {
      "field": "uses",
      "target": "ZC179DD2P4"
    },
    {
      "field": "defines",
      "target": "5SP8E66T9E"
    },
    {
      "field": "about",
      "target": "G003WR6SMX"
    },
    {
      "field": "about",
      "target": "PDFK69ZMST"
    }
  ],
  "body": "\n## Summary\n\nA CW complex is a space built by gluing closed balls of increasing dimension, called cells.\nEach ball is glued along its boundary sphere to the cells already present.\nThe class of CW complexes is flexible enough to contain spheres, projective spaces, Grassmannians, all [[simplicial complexes|V8GZ0W1ZZ5]] and, up to homotopy equivalence, all manifolds.\nIt is rigid enough for homotopy theory to work well.\nWeak homotopy equivalences between CW complexes are homotopy equivalences, and homology can be computed cell by cell.\nThe letters C and W stand for closure-finite and weak topology, the two conditions of J. H. C. Whitehead's definition.\n\n## Explanations\n\n### L1 human\n\nStart with a set of points, the $0$-cells.\nGlue intervals to them by their ends, giving a graph.\nThen glue disks along their boundary circles to the graph, then solid balls along their boundary spheres, and so on.\nThe sphere $S^2$ is a CW complex with one point and one disk whose whole boundary circle is glued to that point.\nThe torus is a CW complex with one point, two loops and one square glued along the word $aba^{-1}b^{-1}$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "57Q05",
      "55U10"
    ],
    "name": "CW complex",
    "same_as": {
      "wikidata": "Q189061",
      "wikipedia": "CW complex"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/CW_complex"
      }
    ],
    "tag": "5W95QFHXPR"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/5W95QFHXPR/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/5W95QFHXPR.md",
  "msc": [
    "57Q05",
    "55U10"
  ],
  "name": "CW complex",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5W9/5W95QFHXPR.md",
  "status": null,
  "tag": "5W95QFHXPR",
  "uri": "m2://5W95QFHXPR",
  "url": "https://math2.tamnd.com/wiki/cw-complex/",
  "writing": {
    "tag": "5W95QFHXPR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}