This discussion focuses on non-translation invariant metrics within the context of metric spaces, specifically targeting beginners in the subject. Two notable examples are presented: the metric on the positive real line defined by d(x,y) = |log(x/y)| and the metric in Euclidean space given by d(x,y) = |x| + |y|, with d(x,x) = 0 for identical points. The conversation emphasizes the importance of understanding these metrics, as they may have applications in both pure and applied mathematics, despite the fact that many metrics exist outside of vector space structures where translation invariance is irrelevant.
PREREQUISITESThis discussion is beneficial for students of mathematics, particularly those studying metric spaces, as well as educators seeking to illustrate complex concepts with relatable examples.