Consider the proof of Proposition 1.11. Suppose that \\(\sim\\) is an equivalence relation, and let \\(P\\) be the set of \\((\sim)\\)-closed and \\((\sim)\\)-connected subsets \\((A_p)_{p \in P}\\).

1. Show that each \\(A_p\\) is nonempty.
2. Show that if \\(p \neq q\\), i.e. if \\(A_p\\) and \\(A_q\\) are not exactly the same set, then \\(A_p \cap A_q = \emptyset\\).
3. Show that \\(A = \bigcup_{p \in P} A_p\\).