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Are compact sets in a geodesic metric space necessarily bounded? What about, if the space is proper?

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What do you mean "proper"? – rschwieb Aug 23 '12 at 12:19
@rschwieb usually a proper metric space or sometimes a heine-borel metric space is a metric space in which every closed ball is compact. – JSchlather Aug 23 '12 at 13:36
@JacobSchlather Thanks :) – rschwieb Aug 23 '12 at 14:06

Compact sets in any metric space are bounded. This is easily seen by taking any point of the set and covering the space with the union of open balls of radius $n$ centered on that point.

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