I am trying to understand this answer. I am trying to understand this answer. In both directions of proof, I am confused why we take a finite subset of the indexing set. Someone seems to have had the same doubt as me and repeated it in the comments, and they got the following reply.
Because a basis for the product topology on $∏_{α∈A}X_α$ consists of sets of the form $∏_{α∈A}U_α$ , where $U_α⊆X_α$ are open sets and only finitely many $U_α$ s are not equal to the corresponding $X_α$ . Considering only basic open sets of this form simplifies the problem
Ok, this makes total sense when we think of it in the product topology, but I am having difficulty thinking of what this means when considering the subset of that, which is done in pointwise convergence topology.