 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

If the inclusion is continuous from $(A,\mathcal T')$, then $i^{-1}(U)=A\cap U$ belongs to $\mathcal T'$ for every $U\in\mathcal T$. These intersections are precisely all members of $\mathcal T_A$, so
$$
\boxed{\mathcal T_A\subseteq\mathcal T'.}
$$
Thus the <subspace topology> is the coarsest <topology> on $A$ making the inclusion continuous.

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