 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2007/e/2/ii/solution

If the inclusion is continuous from

(A, T^{'})

, then

i^{- 1} (U) = A \cap U

belongs to

T^{'}

for every

U \in T

. These intersections are precisely all members of

T_{A}

, so

T_{A} \subseteq T^{'} .

Thus the subspace topology is the coarsest topology on

A

making the inclusion continuous.

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