3 ms·
> Things fall apart when the space is not compact. Your sequences may converge but not to an element in your space. This is about closedness only. ℝ is not co
by blt 3y ago
> Things fall apart when the space is not compact. Your sequences may converge but not to an element in your space.
This is about closedness only. ℝ is not compact but contains its limit points.
- Paul-Craft 3y agoYep. The reason compact spaces are nice is because compactness lets you reduce something that's infinite to something that's finite.
- fiforpg 3y agoIt could be that the author was thinking about sequential compactness: every sequence of elements of the space has a convergent subsequence (with its limit also in the same space). For metric spaces, sequential and usual compactness coincide: https://en.m.wikipedia.org/wiki/Sequentially_compact_space https://en.m.wikipedia.org/wiki/Sequentially_compact_space
- Tainnor 3y agoYes, but not every sequence in R converges. That's of course also true of a compact space, the author was being imprecise there. What they meant was definitely sequential compactness, as mentioned by a sibling comment.