This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general properties of metric spaces? Is every good example just a subset of $\mathbb{R}^n$?"
I will explain. During many proofs, I visualize something like $\mathbb{R}^2$. The professor always draws these pictures:
This is good for many theorems, since much of the analysis is motivate by questions about $\mathbb{R}^n$. But my mental picture is now only $\mathbb{R}^n$. For appreciate the study of metric spaces in full generality, and for intuition, I request more useful examples of metric spaces that are significantly different from $\mathbb{R}^n$, and are not contain in $\mathbb{R}^n$.
Here, I say "useful" to mean that the example "could naturally arise in another context", in mathematics or an application. It is frustrate when I ask why some property does not hold in general, and someone tells me consider the discrete metric. Yes, it is true, and it is easy to see, but the discrete metric is stupid. Does my property fail in any metric space that someone would care about?
In other words, I have the following taxonomy of metric spaces:
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$\mathbb{R}^n$ and the subsets
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degenerate examples like the discrete metric
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contrived examples that I would not see except in analysis (I mean if they are only exist to be pathological, this is where I think to place the Cantor set)
I want to expand this taxonomy, so that when I hear the new definition or theorem, I can compare to this collection of good examples. I know there must be examples with intricate and intuitive interpretations in statistics, science, and engineering.
Which metric spaces are not in my mental taxonomy? In general, what are the useful, non-obvious metric spaces that a student should keep in his mind when learning analysis?
To me, the ideal answer includes the description of the metric space, what properties it has to be unique and different, some consequences of the properties that make different from my examples, and (if not obvious) where I could find the metric space in practice.
One last thing is that I am looking specifically for the examples which are different as metric spaces, so no equivalence to $\mathbb{R}^n$ or any of the subsets or my other list items. I mean isometry with equivalence, I think, but maybe to homeomorphism, I am not sure how I make the best cut. I am just unsatisfy for spaces that look too much like what I could build in $\mathbb{R}^n$ and use in practice.
Best Answer
Nice question. I'm not sure that this will be exactly what you want, but let's go for it: consider $X = \{A,B,C,D\}$, and define $d: X \times X \to \Bbb R$ putting $d(p,p) = 0$ for all $p \in X$, $d(p,q) = d(q,p)$ for all $p,q \in X$, and:
$$d(A,B) = d(A,C) = d(B,C) = 2, \qquad d(A,D) = d(B,D) = d(C,D) = 1.$$ Here, $1$ and $2$ are for simplicity (note: $1 < 2$). Here's a picture:
I like to think that this little space is useful for convincing myself that a result I'm seeing for the first time is true.