| 1. | A metric space is compact iff it is complete and totally bounded.
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| 2. | A metric space is compact if and only if it is complete and totally bounded.
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| 3. | The image of a totally bounded subset under a uniformly continuous function is totally bounded.
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| 4. | The image of a totally bounded subset under a uniformly continuous function is totally bounded.
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| 5. | Therefore both names ( cauchy-precompact and totally bounded ) can be used interchangeably.
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| 6. | The Pacific Ocean is the only ocean which is almost totally bounded by subduction zones.
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| 7. | Further, a uniformity is compact if and only if it is complete and totally bounded.
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| 8. | A uniform space is compact if and only if it is both totally bounded and Cauchy complete.
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| 9. | In fact, a metric space is compact if and only if it is complete and totally bounded.
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| 10. | It turns out that the space is cauchy-precompact if and only if it is totally bounded.
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