{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA pair of spaces $(X, A)$ is a topological space $X$ with a subspace $A$.\nA map of pairs $f \\colon (X, A) \\to (Y, B)$ is a continuous map $f \\colon X \\to Y$ with $f(A) \\subseteq B$.\nWrite $X$ for the pair $(X, \\emptyset)$.\nAn *(ordinary) homology theory* consists of functors $h_n$, for $n \\in \\Z$, from pairs of spaces to [[abelian groups|082GHNPJ53]], and natural transformations $\\partial \\colon h_n(X, A) \\to h_{n-1}(A)$.\nThe *Eilenberg–Steenrod axioms* are the following conditions:\n\n1. (Homotopy) If $f, g \\colon (X, A) \\to (Y, B)$ are [[homotopic|F2F143KQ2M]] through maps of pairs, then $f_* = g_*$.\n2. (Exactness) For each pair, the sequence $\\cdots \\to h_n(A) \\to h_n(X) \\to h_n(X, A) \\xrightarrow{\\partial} h_{n-1}(A) \\to \\cdots$, with the unnamed maps induced by inclusions, is [[exact|JB7ZBFBA9C]].\n3. (Excision) If $Z \\subseteq A$ and the [[closure|0QEAJB91GZ]] of $Z$ lies in the [[interior|0RZH6W98HK]] of $A$, then the inclusion $(X \\setminus Z, A \\setminus Z) \\to (X, A)$ induces isomorphisms $h_n(X \\setminus Z, A \\setminus Z) \\to h_n(X, A)$.\n4. (Dimension) For a one-point space $P$, $h_n(P) = 0$ for all $n \\ne 0$.\n5. (Additivity) If $X$ is the disjoint union of spaces $X_\\alpha$, the inclusions induce an isomorphism $\\bigoplus_\\alpha h_n(X_\\alpha) \\to h_n(X)$.\n\nThe group $h_0(P)$ is the *coefficient group* of the theory.\n\n## Remarks\n\nThe additivity axiom was added later by Milnor; it is automatic for finite disjoint unions.\nA *cohomology theory* is defined dually by contravariant functors $h^n$ and natural maps $\\delta \\colon h^n(A) \\to h^{n+1}(X, A)$, with products in place of direct sums in the additivity axiom.\n\nExample: [[singular homology|M0Q5CFJ7ZH]] with coefficients in an abelian group $G$ satisfies all five axioms, and its coefficient group is $H_0(P; G) = G$.\n\nNon-example: complex topological K-theory, written as a cohomology theory $K^n$, satisfies the first three axioms and additivity for finite unions, but $K^{-2}(P) \\cong \\Z$, so it fails the dimension axiom.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "YHK51GHK2T",
    "kind": "definition",
    "msc": [
      "55N40"
    ],
    "name": "Eilenberg–Steenrod axioms",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms"
      }
    ],
    "tag": "4JG5Y1E5F3",
    "uses": [
      "082GHNPJ53",
      "F2F143KQ2M",
      "JB7ZBFBA9C",
      "0QEAJB91GZ",
      "0RZH6W98HK",
      "M0Q5CFJ7ZH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/4JG5Y1E5F3/",
  "links": [
    {
      "field": "defines",
      "target": "YHK51GHK2T"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    },
    {
      "field": "uses",
      "target": "F2F143KQ2M"
    },
    {
      "field": "uses",
      "target": "JB7ZBFBA9C"
    },
    {
      "field": "uses",
      "target": "0QEAJB91GZ"
    },
    {
      "field": "uses",
      "target": "0RZH6W98HK"
    },
    {
      "field": "uses",
      "target": "M0Q5CFJ7ZH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4JG5Y1E5F3.md",
  "msc": [
    "55N40"
  ],
  "name": "Eilenberg–Steenrod axioms",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4JG/4JG5Y1E5F3.md",
  "status": null,
  "tag": "4JG5Y1E5F3",
  "uri": "m2://4JG5Y1E5F3",
  "url": "https://math2.tamnd.com/wiki/eilenberg-steenrod-axioms-definition/",
  "writing": {
    "tag": "4JG5Y1E5F3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}