Equivalent metrics - Imported Knowlpedia Content

Two equivalent metrics $d$ and $d'$ on the same set $X$ are metrics that induce the same metric-induced topology on $X$ ; equivalently, a set $U\subseteq X$ is open with respect to $d$ if and only if it is open with respect to $d'$ .

Equivalent characterizations

Equivalently, the identity map $\mathrm{id}\colon (X,d)\to (X,d')$ 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 $d$ on $X$ , the metric $d'(x,y)=\min\{1,d(x,y)\}$ is equivalent to $d$ .
On $\mathbb{R}^n$ , the Euclidean metric and the taxicab metric $d_1(x,y)=\sum_{i=1}^n |x_i-y_i|$ are equivalent.