Suppose there are k hypotheses H1,..., Hk to be tested and the overall type I error rate is α.
The closed testing principle allows the rejection of any one of these elementary hypotheses, say Hi, if all possible intersection hypotheses involving Hi can be rejected by using valid local level α tests; the adjusted p-value is the largest among those hypotheses.
It controls the family-wise error rate for all the k hypotheses at level α in the strong sense.
Suppose there are three hypotheses H1,H2, and H3 to be tested and the overall type I error rate is 0.05.
As such, it controls the family-wise error rate for all the k hypotheses at level α in the strong sense.