 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009/e/e4/b/i/solution

Let

A

be clopen and

C

an equivalence class. The intersections

C \cap A

and

C ∖ A

are complementary relatively open subsets of

C

C

means one must be empty. Thus every equivalence class is entirely in

A

or entirely outside it, proving

q^{- 1} (q (A)) = A

. This is precisely saturation.

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