 Solution

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

Every open set of the subspace is

A \cap U

for some

U \in T

. Since

f

takes values in

A

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

The right side is open in

Z

by the assumed continuity of the composition. Hence preimages of every

T_{A}

-open set are open, proving $f : Z \to (A, T_{A})$ is continuous.

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