Let $X$ be a topology space. $\mathbb C[t]$ be the set of polynomial functions. Define the map $$a: X \rightarrow \mathbb C[t], \quad x \mapsto \sum_{i=1}^n a_i(x) t^i $$ where $a_i:X \rightarrow \mathbb C$ for $i=1, \ldots, n$ are maps. :
What topology do we give $\Bbb C[t]$ so that the following holds?
$a$ is continuous if and only if each $a_i$ is continuous.
EDIT: After thinking the comment for some time... I believe this is the topology. We identify $\Bbb C[t]$ as $\Bbb C^{\Bbb N}$. Hence. $a:X \rightarrow \Bbb C[t]$ is continuous iff each $a_i$ is continuous. Please correct me if this is wrong.