Alexander–Spanier cohomology

Definition

Let $X$ be a topological space and let $G$ be an abelian group. For $q \ge 0$, let $C^q(X; G)$ be the group of all functions $f \colon X^{q+1} \to G$. The coboundary $\delta \colon C^q(X; G) \to C^{q+1}(X; G)$ is $$(\delta f)(x_0, \ldots, x_{q+1}) = \sum_{i=0}^{q+1} (-1)^i f(x_0, \ldots, \widehat{x_i}, \ldots, x_{q+1}),$$ where the hat means that $x_i$ is left out. Then $\delta \circ \delta = 0$, so $C^\bullet(X; G)$ is a cochain complex.

A function $f \in C^q(X; G)$ is locally zero if there is an open cover $\mathcal{U}$ of $X$ with the following property. If the points $x_0, \ldots, x_q$ all lie in one member of $\mathcal{U}$, then $f(x_0, \ldots, x_q) = 0$. The locally zero functions form a subcomplex $C^\bullet_0(X; G)$, because $\delta$ maps locally zero functions to locally zero functions. Let $\bar{C}^\bullet(X; G) = C^\bullet(X; G) / C^\bullet_0(X; G)$. The Alexander–Spanier cohomology of $X$ with coefficients in $G$ is the cohomology of this cochain complex: $$\bar{H}^q(X; G) = H^q(\bar{C}^\bullet(X; G)).$$