Definition 20.23.2. Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. Assume given a total ordering on $I$. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}_{ord}^\bullet (\mathcal{U}, \mathcal{F})$ is the ordered Čech complex associated to $\mathcal{F}$, the open covering $\mathcal{U}$ and the given total ordering on $I$.

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