# Total Boundedness is not Preserved under Homeomorphism

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## Theorem

Let $M = \struct {A, d}$ be a totally bounded metric space.

Let $M' = \struct {A', d'}$ be a metric space.

Let $M$ be homeomorphic to $M'$.

Then it is not necessarily the case that $M'$ is totally bounded.

## Proof

## 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