 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2010/e/e3/b/i/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

K_{a} = ⋃ {S \subseteq X : S connected, a \in S}

has exactly the required largest-set property.

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