Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / E / E3 / b / ii / Solution

If $K_a\cap K_b\ne \varnothing$, their union is connected by (a)(iii). It contains $a$, so maximality gives $K_a\cup K_b\subseteq K_a$ and hence $K_b\subseteq K_a$. Interchanging the roles gives the reverse inclusion. Thus the two components coincide; otherwise they are disjoint.