Let $(X_{j},\tau_{j})$ be topological space for $j\in J$ where $J$ is some arbitrary index set. Define the base of the product topology $\tau$ as
$$\mathcal{B}:=\{\times_{j\in J}A_{j}: A_{j}\in \tau_{j}, \text{ and } A_{j}=X_{j} \text{ for all but finitely many }j\}$$
Consider the projections $\operatorname{pr}_{j}:X:=\times_{i \in J} X_{i}\to X_{j}$. My question does not pertain to the part of proving the product topology $\tau$ as the coarsest topology so that the projections are continuous but rather why are the projections necessarily continuous w.r.t. $\tau$?
My thinking:
Consider $A_{j} \in \tau_{j}$, where $j \in J$, clearly $C:=\{ \times_{i \in J}C_{i}:C_{i} = X_{i} \text{ for all but finitely many } i \in J \text{ and }C_{j}=A_{j}\} \subseteq \operatorname{pr}_{j}^{-1}(A_{j})$
and clearly $C\subseteq \mathcal{B}$. But at the same time, the set $D:=\{ \times_{i \in J}D_{i}:D_{i} \neq X_{i} \text{ for infinitely many } i \in J \text{ and }D_{j}=A_{j}\} \subseteq \operatorname{pr}_{j}^{-1}(A_{j})$
$D$ is certainly not a subset of $\mathcal{B}$. How do I know whether $D\subseteq \tau$?
I may be missing a key concept here, it's been a long day, but I'd appreciate any help.
Best Answer
For starters you’re making it much too hard: if $C_j=A_j$, and $C_i=X_i$ for $i\in J\setminus\{j\}$, then $\operatorname{pr}_j^{-1}[A_j]=\prod_{i\in J}C_i\in\tau$. In other words, the inverse image under $\operatorname{pr}$ of an open set in $X_j$ is open in the product, and $\operatorname{pr}$ is therefore continuous. (In fact it’s not just open: it’s a basic open set.)
But what you’ve written also makes no sense: your sets $C$ and $D$ are collections of subsets of $X$ and therefore cannot be subsets of $\operatorname{pr}^{-1}[A_j]$. Individual members of $C$ and $D$ could be subsets of $\operatorname{pr}^{-1}[A_j]$, and indeed some are — $\operatorname{pr}^{-1}[A_j]\in C$, for instance — but that’s a very different matter.
It does, however, make sense to ask whether $D\subseteq\tau$, and the answer is no. In fact $D\cap\tau=\varnothing$: product sets that restrict infinitely many factors are not open in the product topology, because they are not unions of basic open sets in the product.