>>10313208no one really knew whether those were enough to get anything interesting until they studied it, and found a ton of results relating to it.
sure, there are some weird metric spaces, but you can prove a whole lot of interesting stuff about them which makes them rich enough for discussion. so those 3 requirements are okay for now.
if you look more into metric spaces, you'll find that sometimes we impose extra requirements to find out more stuff. for example, sometimes we talk about compact metric spaces, which have an extra requirement. this gives us a lot more to work with, but less objects that we commonly use are "compact." so the stuff we are able to learn about compact metric spaces doesn't apply in general to every metric space.
in fact, there's an even weaker idea of metric spaces that lets us formalize "closeness." topological spaces are an even more general object. every metric space is a topological space, but there are way more topological spaces.
of course, we can figure out even less about topological spaces than we can about metric spaces! but we can still figure out quite a lot.
on the other hand, there is such thing as a normed linear space, which is a metric space that has vector space structure. so we can add and scale things. and even further, there are banach spaces and hilbert spaces. these require even more assumptions. we can learn a ton about these, and they are very rich, but they are also very difficult to study since we have so much to work with. metric spaces are an easier object to grasp at first.
