Definition:Cauchy Equivalent Metrics

Definition

Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$.

That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same underlying set $X$.

Then $d_1$ and $d_2$ are said to be Cauchy equivalent if and only if for every sequence $\sequence {x_n}$ of points in $X$:

$\sequence {x_n}$ is a Cauchy sequence in $\struct {X, d_1} \iff \sequence {x_n}$ is a is a Cauchy sequence in $\struct {X, d_2}$

Source of Name

This entry was named for Augustin Louis Cauchy.

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed fields: Definition $1.9$