Definition 20.9.1. Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ is the Čech complex associated to $\mathcal{F}$ and the open covering $\mathcal{U}$. Its cohomology groups $H^ i(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}))$ are called the Čech cohomology groups associated to $\mathcal{F}$ and the covering $\mathcal{U}$. They are denoted $\check H^ i(\mathcal{U}, \mathcal{F})$.

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