{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nThe *category of topological spaces* $\\mathbf{Top}$ has the [[topological spaces|0CQHFPGJ45]] as objects and the [[continuous maps|0VGFHTM75Y]] as morphisms.\nComposition is the composition of maps, and the identity of $X$ is $\\id_X$.\nThis is a [[category|A2DE66937D]].\nIf $f \\colon X \\to Y$ and $g \\colon Y \\to Z$ are continuous and $U \\subseteq Z$ is open, then $(g \\circ f)^{-1}(U) = f^{-1}(g^{-1}(U))$ is open, so $g \\circ f$ is continuous.\nThe identity map is continuous, and composition of maps is associative.\n\n## Remarks\n\nThe isomorphisms of $\\mathbf{Top}$ are the homeomorphisms.\nA continuous bijection need not be an isomorphism.\nThe identity map from $\\R$ with the discrete topology to $\\R$ with the usual topology is continuous and bijective, but its inverse is not continuous.\n$\\mathbf{Top}$ is a [[complete category|FWGPKXCPB7]] and is also cocomplete: the product is the product topology, and the coproduct is the disjoint union topology.\nThe forgetful functor to sets has a left adjoint (the discrete topology) and a right adjoint (the indiscrete topology).\n$\\mathbf{Top}$ is not [[Cartesian closed|BFBNQ36QS8]]; for this reason algebraic topology often works in the full subcategory of compactly generated spaces.\n\n## Explanations\n\n### L1 human\n\nThe category of topological spaces gathers all spaces and all continuous maps between them.\nIts isomorphisms are the homeomorphisms, the maps that topology regards as \"the same shape\".\nFor example, the map $t \\mapsto (\\cos 2\\pi t, \\sin 2\\pi t)$ from $[0, 1]$ to the circle is a morphism, though not an isomorphism, since it glues the two ends of the interval.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "A2DE66937D"
    ],
    "kind": "example",
    "msc": [
      "18F60",
      "54B30"
    ],
    "name": "Category of topological spaces",
    "same_as": {
      "wikidata": "Q2000555",
      "wikipedia": "Category of topological spaces"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Category_of_topological_spaces"
      }
    ],
    "tag": "K1DX0S4797",
    "uses": [
      "0CQHFPGJ45",
      "0VGFHTM75Y"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/K1DX0S4797/",
  "links": [
    {
      "field": "instance_of",
      "target": "A2DE66937D"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "0VGFHTM75Y"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/K1DX0S4797.md",
  "msc": [
    "18F60",
    "54B30"
  ],
  "name": "Category of topological spaces",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K1D/K1DX0S4797.md",
  "status": null,
  "tag": "K1DX0S4797",
  "uri": "m2://K1DX0S4797",
  "url": "https://tamnd.github.io/math2/wiki/category-of-topological-spaces/",
  "writing": {
    "tag": "K1DX0S4797",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}