If $T_t=(E_t,\tau_t)$ are Polish, $t\in I$, is $T^I:=(\prod_{t\in I}E_t,\prod_{t\in I}\tau_t)$ Polish? ($\prod_{t\in I}\tau_t$ being the product topology.)
If this is not the case, what extra conditions, if any, can be imposed on $T_t$ and $I$ to make $T^I$ Polish? For instance, what if $I\subseteq\mathbb{R}$? What if $E_t$ are all the same? Are countable? What if $\tau_t=\mathbb{P}E_t$? What if all of the above?
A Polish space is a separable, completely metrizable space. The product of uncountably many spaces containing at least two points is not metrizable, so you must impose the condition that at most countably many of the factors are non-trivial. With that condition, the product is Polish: the product of countably many separable spaces is separable, and the product of countably many completely metrizable spaces is completely metrizable.
