Krdytkyu

According to Munkres, the product topology has a subbasis which the union of all $S_i$ such that:

$$S_i ={pi_i^{-1}(U): U text{ is open in }X_i }$$

I am quite certain that I am misunderstanding this definition. To me this looks like $S_i$ contains all Cartesian products such that the i'th element is open in $X_i$. However, that would make the union of all $S_i$ the power set of the product space. It looks to me like Munkres, and other resources as well, are treating this definition as one where $S_i$ comprises all sets of the form

$$dots X_{i-3}times X_{i-2} times X_{i-1} times U times X_{i+1} times X_{i+2} times dots$$

Where U is an open set in $X_i$. All other sets in this product are the entire space $X_j$. Why is this the case? Why, for instance, doesn't $S_i$ contain this set:

$$dots U_{i-3}times U_{i-2} times U_{i-1} times U times U_{i+1} times U_{i+2} times dots$$

Where $U_j$ may or may not be open in $X_j$ for $j neq i$?

You are misunderstanding $pi_i^{-1}(U)$. This is the set which has $X_j$ in the $j^{th}$ slot (for $jneq i$) and $U$ in the $i^{th}$ slot.

For example, in $mathbb R^3$, one has $pi_1^{-1}(A) = Atimes{mathbb R}times{mathbb R}$.

In other words, $pi_i^{-1}(U)$ is the single set of all points in the product which project into $U$ under the action of $pi_i$.

But your first formula is exactly what $pi_i^{-1}[U]$ means: the only condition for a point to be in that set, is that the $i$-th coordinate of that point is in $U$; all other coordinates are completely free.

It contains the other set as a subset, but topologies aren't closed under subsets, so that doesn't mean anything. It can be written as $bigcap_{i in I} pi_i^{-1}[U_i]$, but that will be an infinite intersection, and topologies are only guaranteed to be closed under finite intersections.