Acyclic open cover for a sheaf

Assume the Axiom of Choice, let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X$, with sheaf cohomology $H^q(W,\mathcal{F}{\mathrm{\mid }}_W)$ of an open subspace as in Sheaf cohomology as right derived global sections and acyclicity over an open subspace as in Gamma-acyclic abelian sheaf; the Axiom of Choice is assumed because these groups are formed from the supplied injective resolutions (The Axiom of Choice).

Let $\mathcal{U}=(U_i{)}_{i\in I}$ be an open cover of $X$ indexed by a linearly ordered set, with ordered Čech cochains $C^{\bullet }(\mathcal{U},\mathcal{F})$ (Ordered Čech cochain complex of a cover). The cover $\mathcal{U}$ is $\mathcal{F}$-acyclic, or a Leray cover for $\mathcal{F}$, when for every nonempty finite set of indices $\{i_0,\dots ,i_p\}\subseteq I$ the sheaf $\mathcal{F}$ is ${\mathrm{\Gamma }}_W$-acyclic on the open subspace $W:=U_{i_0}\cap \cdots \cap U_{i_p}$, that is, when $H^q(W,\mathcal{F}{\mathrm{\mid }}_W)=0\text{for every }q>0,$ where $\mathcal{F}{\mathrm{\mid }}_W$ is the restriction of $\mathcal{F}$ to the open subspace $W$ (Restriction of a sheaf to an open subspace, Gamma-acyclic abelian sheaf).

Equivalently: $\mathcal{F}$ is ${\mathrm{\Gamma }}_W$-acyclic for every open set $W\subseteq X$ that is a finite nonempty intersection of members of $\mathcal{U}$. The single member case $p=0$ is included, so every member $U_i$ is required to be such a $W$, and the condition involves only the open sets occurring in $\mathcal{U}$ and not the chosen linear order of the index set: reindexing $\mathcal{U}$, or passing to a cover with the same members, does not change whether $\mathcal{U}$ is $\mathcal{F}$-acyclic. An intersection $U_{i_0}\cap \cdots \cap U_{i_p}$ that happens to be empty contributes no condition: the space $W$ is then empty and every sheaf on it is the zero sheaf, because $\mathcal{G}(\mathrm{\varnothing })$ is a singleton (A set-valued sheaf has a unique section over the empty open set), so $\mathrm{\Gamma }(W,-)$ is the zero functor, which is exact, and its positive derived functors vanish (An exact functor has vanishing positive derived functors).