In Real
Let
- The set
is closed and bounded - every sequence
has a convergent subsequence .
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