1. A metric space is Lindel�f if and only if it is second-countable. 2. Every metric space is therefore, in a natural way, a topological space. 3. A metric space is compact iff it is complete and totally bounded. 4. Every isometry group of a metric space is a subgroup of isometries. 5. For example, a metric space can be regarded as an enriched category. 6. The definition can be generalized to functions that map between metric spaces . 7. In general, a metric space may have no geodesics, except constant curves. 8. The space has a metric ( see metric space for details ). 9. Thus for metric spaces we have : compactness = cauchy-precompactness + completeness. 10. In a general metric space , however, a Cauchy sequence need not converge.
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