 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2008/e/4/c/solution

The inclusion

f (⋂_{n} K_{n}) \subseteq ⋂_{n} f (K_{n})

holds for every map. For the reverse choose

y \in ⋂_{n} f (K_{n})

and let

L_{n} = K_{n} \cap f^{- 1} ({y})

. These sets are nonempty and nested. A Hausdorff space has closed singletons, and continuity makes the fibre closed, so each

L_{n}

is compact. Part (b)(i) gives a point

x \in ⋂_{n} L_{n}

. Then

x \in ⋂_{n} K_{n}

and

f (x) = y

. Hence

f (n ⋂ K_{n}) = n ⋂ f (K_{n}) .

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