Intersection hypothesis ($H_I={\bigcap }_{i\in I}{H}_i$)

For a nonempty finite index set $I$, this hypothesis asserts that every member $H_i$ is true simultaneously. A valid local test must have its stated size under every parameter satisfying the entire intersection. The Bonferroni correction within $I$ always supplies such a test from valid elementary p-values.

The closure is the collection of all nonempty intersection hypotheses from the original elementary family, including singletons. It is not a topological closure. The closed testing procedure requires local level-$\alpha$ tests for these intersections and rejects an elementary hypothesis only when all intersections containing it reject.