Equivalence of Metrics is not Defined/Mistake

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Source Work

1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.):

Part $\text I$: Basic Definitions
Section $5$. Metric Spaces
Complete Metric Spaces


The concept of equivalence of metrics is not defined, although the concept is mentioned and used in the context of complete metric spaces.

Equivalent Metrics

The definition is as follows:

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 set $X$.

Let $\sequence {x_n}$ be a sequence in $X$.

Let $n \to \infty$.

Suppose that $x_n \to x$ in $\struct {X, d_1}$ if and only if $x_n \to x$ in $\struct {X, d_2}$.

Then $d_1$ and $d_2$ are equivalent metrics.