Metric Space Compact iff Complete in All Equivalent Metrics
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Theorem
Let $M_1 = \struct {A, d_1}$ be a metric space.
Then $M_1$ is compact if and only if $M_2 = \struct {A, d_2}$ is a complete metric space whenever $d_2$ is equivalent to $d_1$.
Proof
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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