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Suppose a metric space has a bounded metric and it has a Heine-Borel property. So each set is bounded, is any closed set then is compact?

Richard Kim
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1 Answers1

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Yes it is. That is what Heine-Borel property says.

Note that in this case the whole space $X$ itself if compact.

Kavi Rama Murthy
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  • Thanks, but what about the case when the whole space itself is not compact. – Richard Kim Dec 14 '19 at 12:05
  • @RichardKim You are putting conditions that make $X$ compact. You are assuming that $X$ is bounded and it is always closed so H-B property implies that $X$ is compact. If a metric space is not compact the either it does not have H -B property or the metric is not bounded. – Kavi Rama Murthy Dec 14 '19 at 12:07
  • There are infinite-dimensional spaces that have H -B property and they are metrizable, eg. nuclear Frechet mnaifolds – Richard Kim Dec 14 '19 at 12:10
  • Yes, but the metrics on them are not bounded. @RichardKim – Kavi Rama Murthy Dec 14 '19 at 12:12
  • Metrics are bounded, indeed. Furthermore, each metric space has an equivalent bounded metric – Richard Kim Dec 14 '19 at 12:16
  • @RichardKim Heine Borel peoperty refers to a particular metric. You cannot. change the metric to an equivalent metric and use H-B property. – Kavi Rama Murthy Dec 14 '19 at 12:17