 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2010/e/1/b/iii/solution

We prove (iii)

\Rightarrow

(i). If

C = \overline{B} ∖ B

is closed in

X

U = X ∖ C

and

F = \overline{B}

. Then

U

is open,

F

is closed, and

U \cap F = (X ∖ (\overline{B} ∖ B)) \cap \overline{B} = B .

Together the three implications prove the equivalence. These are the standard characterizations of a locally closed subset.

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