Equivalent Metrics: Convergent Sequences?

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SUMMARY

Two metrics d_1 and d_2 on a set X are equivalent if and only if they have the same convergent sequences. This conclusion is supported by Engelking's "General Topology", specifically Theorem 4.1.2, which states that two metrics are equivalent if they induce the same convergence. The terminology "topologically equivalent" is preferred, as metrics can also be classified as "uniformly equivalent" or "Lipschitz equivalent".

PREREQUISITES
  • Understanding of metric spaces
  • Familiarity with convergence concepts in topology
  • Knowledge of Engelking's "General Topology"
  • Basic concepts of uniform and Lipschitz equivalence
NEXT STEPS
  • Study Engelking's "General Topology" for a deeper understanding of metric equivalence
  • Explore the definitions and implications of uniform equivalence in metric spaces
  • Research Lipschitz equivalence and its applications in analysis
  • Investigate examples of convergent sequences in different metrics
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Mathematicians, students of topology, and anyone studying metric spaces and their properties will benefit from this discussion.

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Is it true that two metrics d_1 and d_2 on a set X are equivalent iff they have the same convergent sequences (i.e., a sequence that converges in d_1 converges in d_2 and vice versa)?
 
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Yes.

Engelking, "General Topology", Thm. 4.1.2

"Two metrics on X are equivalent iff they induce the same convergence."
 
Yup, although the correct term is "topologically equivalent", since metrics can also be "uniformly equivalent" or "Lipschitz equivalent" and there might be more such equivalencies.
 

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