Equivalent metrics

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 $d'$.

Equivalently, the identity map $\mathrm{id}\colon (X,d)\to (X,d')$ is a homeomorphism . 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.