I am currently studying topology with Munkres and I am asking for some general proof techniques and clarification rather than just posting my question and wait for a solution. I am given that the real line is endowed with the euclidean topology and also given a subset of the real line. My task is to show that this subset is open with respect to the box topology, but not open with respect to the product topology.
My question is what is different between the box and product topologies such that a set would be open in the box topology and not open in the product topology? I am familiar with the different bases used to generate each topology, but I am lost as to how the same set could be open with respect to the box topology but not open with respect to the product topology
For $n\in\Bbb N$ let $X_n=\Bbb R$, and let $X=\prod_{n\in\Bbb N}X_n$. For each $n\in\Bbb N$ let $U_n=(0,1)$, the open unit interval, and let $U=\prod_{n\in\Bbb N}U_n$. By definition $U$ is open in the box topology on $X$: it’s even a basic open set.
It is not open in the product topology, however; in fact, it doesn’t even contain any non-empty open set in the product topology. Remember, the usual base for the product topology is the family of all sets of the form $\prod_{n\in\Bbb N}V_n$ such that each $V_n$ is open in $X_n$, and $\{n\in\Bbb N:V_n\ne X_n\}$ is finite. In other words, all but finitely many of the sets $V_n$ are all of $X_n$. If $U$ were open, it would contain one of these sets, but in fact it does not.
To see this, observe that if $\prod_{n\in\Bbb N}V_n\subseteq\prod_{n\in\Bbb N}U_n$, then $V_n\subseteq U_n$ for each $n\in\Bbb N$. If $\prod_{n\in\Bbb N}V_n$ is a basic open set in the product topology, then $V_n=X_n=\Bbb R$ for all but finitely many $n\in\Bbb N$. Pick any $n\in\Bbb N$ such that $V_n=X_n$; then $$V_n=\Bbb R\nsubseteq(0,1)=U_n\,,$$ so $\prod_{n\in\Bbb N}V_n\nsubseteq\prod_{n\in\Bbb N}U_n$.
To put it informally, basic open sets in the box topology can restrict every coordinate, while basic open sets in the product topology can restrict only finitely many coordinates. Thus, you can get basic open sets in the box topology that don’t contain any basic open set in the product topology and therefore are not open in the product topology (and indeed have empty interior).