I have a simple question that comes from Munkres section 19, Example 2.
Let $f:\mathbb{R}\rightarrow\mathbb{R}^{\omega}$ be given by $f(x)=(x,x,x,...)$, with $\mathbb{R}^{\omega}$ a countably infinite product of $\mathbb{R}$.
Munkres goes on to argue that this function is continuous in the product topology but not in the box topology.
For the box topology, consider the basis element $$B=(-1,1)\times\left(-\frac{1}{2},\frac{1}{2}\right)\times\left(-\frac{1}{3},\frac{1}{3}\right)\times...$$
Munkres goes on to argue that $f^{-1}(B)$ is closed. My question is, how is this inverse defined?
If you take the naive definition that $y\in f^{-1}(B) \implies f(y)\in B_{a}$ for some $a$ where $a$ labels the countable elements of $B$, then you have $f^{-1}(B)=(0,1)$.
I feel like I am missing something obvious in all of this.