Past exam of the options II course of the University of Oxford / 2010 / E / E3 / b / i / Solution

LetThis 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 $a$.