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, $X$ will denote a non-empty set and $d_{1}$ and $d_{2}$ will denote two metrics on $X$.
(1) The two metrics $d_{1}$ and $d_{2}$ are said to be topologically equivalent if they generate the same topology on $X$.
(2) Two metrics $d_{1}$ and $d_{2}$ are strongly equivalent if and only if there exist positive constants $\alpha $ and $\beta$ such that, for every $x,\ y\in X$, $$ \alpha d_{1}(x,y) \leq d_{2}(x,y) \leq \beta d_{1} (x, y) $$
