Equivalence of metrics

In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being 'the same', or equivalent. In the study of metric spaces in mathematics, there are various notions of two metrics on the same underlying space being 'the same', or equivalent. In the following, X {displaystyle X} will denote a non-empty set and d 1 {displaystyle d_{1}} and d 2 {displaystyle d_{2}} will denote two metrics on X {displaystyle X} . The two metrics d 1 {displaystyle d_{1}} and d 2 {displaystyle d_{2}} are said to be topologically equivalent if they generate the same topology on X {displaystyle X} . The adjective 'topological' is often dropped. There are multiple ways of expressing this condition: