I'm looking for an undergraduate course in mathematics that would be likely to cover material like this. I suspect maybe this approach to math is called Metric Spaces, but I'm not sure.
It seems to come up a lot in computer science and I'd like to explore the subject more rigorously.
The basics of metric spaces are typically covered in a first course in real analysis, which might be a course titled something like "Real Analysis", "Introduction to Real Analysis", "Introduction to Analysis", or occasionally "Advanced Calculus". They would also be covered in a first course in general topology, which might be titled something like "Introduction to Topology".
To provide some distinction between these two courses, a real analysis course would focus on rigorously justifying and generalizing many ideas from calculus (with metric spaces being one important setting in which to generalize such ideas). A topology course would focus on generalizing a certain special core of these ideas to a much more abstract and general setting called topological spaces (with metric spaces being a special case of topological spaces), possibly with some applications to some more geometric problems (the beginnings of algebraic topology).