 Solution

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

Y

T_{Y} = {Y \cap U : U \in T} .

Y

is open in

X

, each such intersection is open in

X

. Conversely, an open subset of

X

contained in

Y

is its own intersection with

Y

. Thus the open sets of $Y$ are exactly the open subsets of $X$ contained in $Y$ in this case.

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