Are compact sets in a geodesic metric space necessarily bounded? What about, if the space is proper?
Compact sets in geodesic metric space
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$\begingroup$
geometry
metric-spaces
compactness
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0What do you mean "proper"? – 2012-08-23
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0@rschwieb usually a proper metric space or sometimes a heine-borel metric space is a metric space in which every closed ball is compact. – 2012-08-23
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0@JacobSchlather Thanks :) – 2012-08-23
1 Answers
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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.