{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "FVGB8ZP67B"
    },
    {
      "field": "defines",
      "target": "GRZKVSMJXF"
    }
  ],
  "body": "\n## Summary\n\nTake a set and a family of maps from it into topological spaces.\nThe initial topology on the set is the coarsest topology that makes all these maps continuous.\nThe [[subspace topology|0WS9P12FQR]] and the [[product topology|0GBG8B49AA]] are both initial topologies.\nIt is characterised by a universal property: a map into the set is continuous exactly when its composites with all the given maps are continuous.\nThe dual notion is the [[final topology|6ERJRRZK32]].\n\n## Explanations\n\n### L1 human\n\nSuppose you have a set and some maps from it into spaces you already understand.\nYou want the fewest open sets that still make every map continuous.\nSo you take the preimages of open sets under the maps, and only add what you must to get a topology.\nFor example, the projections $\\R^2 \\to \\R$ onto the two coordinates give the usual topology of the plane.\nOpen rectangles are intersections of two such preimages.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "54A10",
      "54B30"
    ],
    "name": "Initial topology",
    "same_as": {
      "wikidata": "Q13416298",
      "wikipedia": "Initial topology"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Initial_topology"
      }
    ],
    "specializes": [
      "0CQHFPGJ45"
    ],
    "tag": "1NWC3710WW"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/1NWC3710WW/",
  "links": [
    {
      "field": "specializes",
      "target": "0CQHFPGJ45"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1NWC3710WW.md",
  "msc": [
    "54A10",
    "54B30"
  ],
  "name": "Initial topology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1NW/1NWC3710WW.md",
  "status": null,
  "tag": "1NWC3710WW",
  "uri": "m2://1NWC3710WW",
  "url": "https://math2.tamnd.com/wiki/initial-topology/",
  "writing": {
    "tag": "1NWC3710WW",
    "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
      }
    ]
  }
}