 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2007/e/e2/iii/solution

A \cap U

is subspace-open, then

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

The right side is open in

Z

by the assumed continuity of the composition. This holds for every subspace-open set, proving $f$ is continuous into the subspace topology.

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