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