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  "body": "\n## Definition\n\nConsider pairs of spaces $(X, A)$ and maps of pairs, as in the [[Eilenberg–Steenrod axioms|YHK51GHK2T]], and write $X$ for $(X, \\emptyset)$.\nA *generalized cohomology theory* consists of contravariant functors $h^n$, for $n \\in \\Z$, from pairs of spaces to [[abelian groups|082GHNPJ53]], and natural transformations $\\delta \\colon h^n(A) \\to h^{n+1}(X, A)$, such that:\n\n1. (Homotopy) [[Homotopic|F2F143KQ2M]] maps of pairs induce the same homomorphisms.\n2. (Exactness) For each pair, the sequence $\\cdots \\to h^n(X, A) \\to h^n(X) \\to h^n(A) \\xrightarrow{\\delta} h^{n+1}(X, A) \\to \\cdots$, with the unnamed maps induced by inclusions, is [[exact|JB7ZBFBA9C]].\n3. (Excision) If $Z \\subseteq A$ and the closure of $Z$ lies in the interior of $A$, then the inclusion $(X \\setminus Z, A \\setminus Z) \\to (X, A)$ induces isomorphisms $h^n(X, A) \\to h^n(X \\setminus Z, A \\setminus Z)$.\n4. (Additivity) If $X$ is the disjoint union of spaces $X_\\alpha$, the inclusions induce an isomorphism $h^n(X) \\to \\prod_\\alpha h^n(X_\\alpha)$.\n\nNo condition is put on the groups $h^n(P)$ of a one-point space $P$.\n\n## Remarks\n\nOften the axioms are only asked for CW pairs, or for pairs of spaces with the homotopy type of CW pairs.\nA *generalized homology theory* is defined in the same way with covariant functors.\n\nExample: [[singular cohomology|6N7BC70CX0]] $H^n(-; G)$ with coefficients in an abelian group $G$ is a generalized cohomology theory, and it also satisfies the dimension axiom $H^n(P; G) = 0$ for $n \\ne 0$.\n\nExample: complex topological K-theory is a generalized cohomology theory with $K^n(P) \\cong \\Z$ for $n$ even and $K^n(P) = 0$ for $n$ odd, by Bott periodicity; it is not an ordinary theory.\n\nNon-example: the homotopy groups $\\pi_n(X, A, x_0)$ of pointed pairs satisfy a homotopy axiom and have long exact sequences, but they fail excision, so they do not form a generalized homology theory.\n",
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