I have asked and read similar questions, but I am still somewhat confused on the notion of "a set being open with respect to some topology". My task is concerning the box topology and the product topology on the infinite product space of the real line.
The goal is to show that a subset (call this subset S) of the infinite product space is open with respect to the box topology but not open with respect to the product topology. For more information, the subset S is a set of infinite-tuples of elements in the real line (ie just numbers). In each tuple, each succeeding tuple element is greater than the previous one. An example of an element of S would be (1, 2, 3, 4, ....) or (1.01, 1.02, 1.03, ...). I imagine that to prove this, we will need to use the slight difference in the bases that generate each topology.
My question is: how can we leverage the basis for each topology to show that some subset of the infinite product space is open/not open with respect to that topology? Put in other words: If we consider just the box topology, how does a subset S have to "relate" to the basis that generates the box topology in order for the subset S to be considered open in the box topology.
My original thought was to show that all elements of S can be represented as the union or finite intersection of base elements for the box topology, but the same does not go for the product topology. I am not sure how I would prove this exactly, or if that is even the correct procedure to follow.
Finally, I want to emphasize I am not looking for a clear cut answer, but to be pointed in the right direction for this proof.
Tl;dr How can we classify a subset of the infinite product space of the real line as open/not open with respect to the box topology/product topology.
HINT: So
$$S=\left\{\langle x_n:n\in\Bbb N\rangle\in\Bbb R^{\Bbb N}:x_n<x_{n+1}\text{ for all }n\in\Bbb N\right\}\,.$$
To show that $S$ is open in the box topology, let $\langle x_n:n\in\Bbb N\rangle\in S$, and try to find open intervals $U_n$ around each $x_n$ such that $\prod_{n\in\Bbb N}U_n\subseteq S$.
To show that $S$ is not open in the product topology, show that no basic open set in the product topology is a subset of $S$; i.e., show that every basic open set in the product topology contains a sequence that is not strictly increasing.