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
- Section $5$. Metric Spaces
Mistake
- 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.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces