I was wondering if anyone had information that would help me better understand the difference, so that I can work better on: Interesting Metrics
I took a look at Metric assuming the value infinity when searching for this, but as I don't have a background yet in this material, I was hoping someone could explain it a bit more simply. To be clear, I don't know the notation too well, and would be happy with a not-terribly technical understanding.
$\textbf{I understand the definitions, but what sorts of properties do we gain and lose?}$ What types of theorems are there that one has but the others don't. Basically, why should I choose to talk about one rather than the other? How much more powerful is metric than pseudo-metric?
Thank you very much.
Whenever I have seen a pseudometric space brought up, the structure is typically not considered interesting in its own right. The main difference between this and a metric space proper is that we cannot use the distance function to distinguish between different points in our space. What usually happens is that next is that we introduce an equivalence relation where $x \sim y$ if $d(x,y) = 0$. This is done, for example, with the $\mathcal{L}^p$-spaces, where we declare functions who differ by a set of measure zero to be equivalent. "Mod" this equivalence relation, then, we now have a proper metric space where we can distinguish between equivalence classes.
So, in summary, the main point of pseudometric spaces is that we cannot use our concept of distance to distinguish between different points, forcing us to think of things in terms of equivalence classes where points declared to have zero distance are considered equivalent.