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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $G$ be an [[abelian group|082GHNPJ53]].\nFor $q \\ge 0$, let $C^q(X; G)$ be the group of all functions $f \\colon X^{q+1} \\to G$.\nThe coboundary $\\delta \\colon C^q(X; G) \\to C^{q+1}(X; G)$ is\n$$(\\delta f)(x_0, \\ldots, x_{q+1}) = \\sum_{i=0}^{q+1} (-1)^i f(x_0, \\ldots, \\widehat{x_i}, \\ldots, x_{q+1}),$$\nwhere the hat means that $x_i$ is left out.\nThen $\\delta \\circ \\delta = 0$, so $C^\\bullet(X; G)$ is a cochain complex.\n\nA function $f \\in C^q(X; G)$ is *locally zero* if there is an open cover $\\mathcal{U}$ of $X$ with the following property.\nIf the points $x_0, \\ldots, x_q$ all lie in one member of $\\mathcal{U}$, then $f(x_0, \\ldots, x_q) = 0$.\nThe locally zero functions form a subcomplex $C^\\bullet_0(X; G)$, because $\\delta$ maps locally zero functions to locally zero functions.\nLet $\\bar{C}^\\bullet(X; G) = C^\\bullet(X; G) / C^\\bullet_0(X; G)$.\nThe *Alexander–Spanier cohomology* of $X$ with coefficients in $G$ is the cohomology of this cochain complex:\n$$\\bar{H}^q(X; G) = H^q(\\bar{C}^\\bullet(X; G)).$$\n\n## Remarks\n\nExample: let $X$ be a single point.\nThen $C^q(X; G) \\cong G$ for every $q$, and no nonzero function is locally zero.\nThe coboundary $C^q \\to C^{q+1}$ is multiplication by $\\sum_{i=0}^{q+1} (-1)^i$, which is $0$ for even $q$ and $1$ for odd $q$.\nSo $\\bar{H}^0(X; G) \\cong G$ and $\\bar{H}^q(X; G) = 0$ for $q \u003e 0$.\n\nNon-example: a function on $X^{q+1}$ that is nonzero at some diagonal point $(x, \\ldots, x)$ is not locally zero, since every cover has a member that contains $x$.\n\nThere is a relative version for a pair $(X, A)$ and a version with compact supports.\nOn paracompact Hausdorff spaces, $\\bar{H}^q(X; G)$ is isomorphic to Čech cohomology and to the sheaf cohomology of $X$ with coefficients in the constant sheaf $G$.\n",
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