Solution
= Solution
Let
$$
\boxed{K_a=\bigcup\{S\subseteq X:S\text{ is connected and }a\in S\}.}
$$
This family is nonempty because it contains $\{a\}$. All its members intersect at $a$, so part (a)(iii) proves that $K_a$ is connected. It contains $a$ and, by its construction, contains every connected subset containing $a$. Thus it has exactly the required largest-subset property. This is the <connected component> containing $a$.