For exercise 1.12.3, I'm somewhat confused about how to go about proving this (proofs were never my strong suit). Is it not possible that

$$\bigcup\limits_{p\in P} A_p = A \cap A_q$$

where $$q \notin P$$

I.e. if you union all the partitions, there might be elements of the set in non-closed and non-connected groups?