Solution
= Solution
Let $K_a$ be the union of all <connected subsets> containing $a$. This collection is nonempty because it includes $\{a\}$. Any two of its members meet at $a$, so (a)(iii) makes their full union <connected>. It contains $a$ and by its definition contains every <connected subset> through $a$. Thus
$$
\boxed{K_a=\bigcup\{S\subseteq X:S\text{ connected},\ a\in S\}}
$$
has exactly the required largest-set property.