  The closure and the partitioning principles have been used to build various
multiple testing procedures in the past three decades. The essence of these two
principles is based on parameter space partitioning. In this article, we
propose a novel approach coined the covering principle from the perspective of
rejection region coverage in the sample space. The covering principle divides
the whole family of null hypotheses into a few overlapped sub-families when
there is a priority of making decisions for hypothesis testing. We have proven
that the multiple testing procedure constructed by the covering principle
strongly controls the familywise error rate as long as the multiple tests for
each sub-familiy strongly control the type I error. We have illustrated the
covering principle can be applied to solve the general gate-keeping problems.
