Intuition behind $D(\mathbf x,\mathbf y)=\sup\left\{\frac{\bar d(x_i,y_i)}{i} \right\}$ inducing the product topology on $\mathbb R^\omega$

Question

Theorem $20.5$ in Munkres' Topology says that $$D(\mathbf x,\mathbf y)=\sup\left\{\frac{\bar d(x_i,y_i)}{i} \right\}$$ induces the product topology on $\mathbb R^\omega$, where $\bar d(a,b)=\min\{1,|a-b|\}$ is the standard bounded metric on $\mathbb R$.

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I understand the proof in the book (it also has some variants on MSE), and I was also able to come up with a proof of my own using Lemma $13.3$ of the book. Could someone help me understand the intuition behind

1. Why one would define $D$ the way it is? It seems a little out of the blue. I would certainly not think of crazy metrics like this unless explicitly told they exist and are useful.
2. Why does $D$ induce the product topology on $\mathbb R^\omega$? I don't want a proof, I already know that - I want to know why one might expect (intuitively) that $D$ induces the product topology on $\mathbb R^\omega$. Pictures, visualizations, etc. might help.

Thank you!

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This post is related, but does not answer my question.

Answer 1

The term $\frac{1}{i}$ in the term we take a sup of (which could also have been $2^{-i}$ or some such decreasing term) ensures that the tail end of the sequences we take the distance of gets less important: if we want to have the distance between $(x_i)_i$ and $y_i)_i$ to be smaller than some $\delta$, say, we know that in the sup all terms $x_i, y_i$ with $i$ from a certain $N$ onwards are irrelevant (as the $\frac{1}{i}$ term plus the truncation in $\bar{d}$ ensure that $\frac{\bar{d}(x_i, y_i)}{i} < \delta$ if $i$ is large enough, so cannot cause the total $D$ to be larger than $\delta$). The same holds for another metric that induces the product topology: $D(\mathbf{x},\mathbf{y}) = \sum_{i=1}^\infty 2^{-i} \bar{d}(x_i, y_i)$, where the series is convergent so the tail end is smaller than any prescribed limit too. So these metrics really depend on the first coordinates mostly, like the product topology which has open basic sets of the form $O_1 \times O_2 \ldots \times O_N \times X_{N+1} \times X_{N+2} \times \ldots$ for all finite indices $N$ and open sets $O_i$. (this corresponds to finite intersections of subbasic open sets $p_i^{-1}[O_i]$, beyond the largest used index we have factors $X_i$ etc.)

In the sup metric proper all coordinates are equally contributing to the metric and so we get different open sets (more of them, but fewer than the box topology).