 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Every open set of the subspace is $A\cap U$ for some $U\in\mathcal T$. 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 $\mathcal T_A$-open set are open, proving \b[$f:Z\to(A,\mathcal T_A)$ is continuous].

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