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\$$.