 Solution

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

Let

K_{a} = ⋃ {S \subseteq X : S 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

 New to topics? Read the docs here!