> 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.
This is about closedness only. ℝ is not compact but contains its limit points.
For metric spaces, sequential and usual compactness coincide:
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.