In the infinite product topology, rigorously prove a given set is not open For example, how do we prove that $(-1, 1)^\omega$ is not open in $\mathbb{R}^\omega$? I understand the general form of an open set in this topology, and it's difference with the box topology. The problem is that, while it seems intuitively evident, I'm not sure how to rigorously and formally prove that such set is not open in the product topology. Furthermore, is it necessary to us reductio ad absurdum to prove it?

You can directly show its complement is not closed. Let $x_n = (0,0,\dots,0,0,2,2,2,2,2,\dots)$ where there are $n$ $0$'s. Then each $x_n \in \mathbb{R}^\omega\setminus (-1,1)^\omega$ but $x_n \to (0,0,0,\dots) \in (-1,1)^\omega$.