 Solution

Suppose two components $K$ and $L$ intersect. Their union is connected by part (a)(iii). It contains the point used to define $K$, so maximality forces $K\cup L\subseteq K$, giving $L\subseteq K$. The same reasoning with $L$ gives $K\subseteq L$. Hence $K=L$. Thus distinct connected components are disjoint.