Equivalent metrics - Knowlpedia

Two metrics $d$ and $d'$ on a set $X$ are equivalent if they induce the same topology on $X$ .

Equivalent characterizations

Equivalently, $\operatorname{id}_X:(X,d)\to(X,d')$ is a homeomorphism.

Remarks

Equivalent metrics have the same open sets and therefore the same convergent sequences, but they may differ in which sequences are Cauchy and whether the space is complete.

Examples

For any metric $d$ , the bounded metric $d'(x,y)=\min\{1,d(x,y)\}$ is equivalent to $d$ .
On $\mathbb R^n$ , the Euclidean and taxicab metrics are equivalent.