Testing hypercompleteness with hypercovers

Theorem 6.40 is often useful in practice: to show that a topos \(T\) is hypercomplete, it suffices to verify that all hypercovers are effective. For sheaf topoi \(\Shv(X)\) on a topological space \(X\), one may work with open hypercovers when these are sufficiently plentiful. Ordinary open covers alone do not detect hypercompleteness, since their Čech nerves are effective in every topos. The dimension criteria of Section 6.5 provide more concrete sufficient conditions.