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 $U\in \mathcal{T}$. Since $f$ takes values in $A$,The right side is open in $Z$ by the assumed continuity of the composition. Hence preimages of every ${\mathcal{T}}_A$-open set are open, proving $f:Z\to (A,{\mathcal{T}}_A)$ is continuous.