Fixed-cover Čech cohomology

Let $X$ be a topological space, $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{U}=(U_i{)}_{i\in I}$ an open cover indexed by a linearly ordered set, with ordered Čech cochain complex $(C^{\bullet }(\mathcal{U},\mathcal{F}),{\delta }^{\bullet })$ (Ordered Čech cochain complex of a cover, The Čech differential squares to zero). The Čech cohomology of the fixed cover $\mathcal{U}$ with values in $\mathcal{F}$ is ${\check{H}}^p(\mathcal{U},\mathcal{F}):=\ker ({\delta }^p:C^p(\mathcal{U},\mathcal{F})\to C^{p+1}(\mathcal{U},\mathcal{F}))/\mathrm{im}({\delta }^{p-1}:C^{p-1}(\mathcal{U},\mathcal{F})\to C^p(\mathcal{U},\mathcal{F})),$ the $p$-th cohomology group of the cochain complex $(C^{\bullet }(\mathcal{U},\mathcal{F}),{\delta }^{\bullet })$ in the sense of Cohomology object of a cochain complex. Since $C^{-1}(\mathcal{U},\mathcal{F})=0$ and ${\delta }^{-1}=0$, one has ${\check{H}}^0(\mathcal{U},\mathcal{F})=\ker ({\delta }^0)$, and for $p<0$ the group ${\check{H}}^p(\mathcal{U},\mathcal{F})$ is $0$ because $C^p=0$, so both the kernel in degree $p$ and the image in degree $p$ are zero. A cochain in $\ker ({\delta }^p)$ is called a Čech $p$-cocycle and a cochain in $\mathrm{im}({\delta }^{p-1})$ a Čech $p$-coboundary.