analysistopology

In Real

Let be a subset of . Then the following two statements are equivalent:

In Metric Space

  • compact sets are both closed and bounded.
  • Let be a Euclidean space with either the Euclidean metric, the texicab metric, or the sup norm metric. Let be a subset of . Then is compact iff it is closed and bounded.

However, the Heine-Borel theorem is not true for general metrics. For instance, the integers with the discrete metric is closed and bounded, but not compact, since the sequence has no convergent subsequence.