Notation of infinite product of banach algebras

Question

Here they define product for countable many. But how to define the that product for uncountable many banach algebras. I am unable to find it anywhere that's why I am posting it here. Below I found a excercise where it does not mention about the index that's why I am wondering.

Answer 1

It can defined by: $\{x\in\prod_{i\in I}\mathcal A_i: \forall \epsilon>0, \{i: \|x(i)\|>\epsilon\} \text{ is finite } \}$

Or we may consider $I$ as a discrete topological space, and $x$ are those sections that vanish at infinity.