Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I read that in a metric space compactness and sequential compactness mean the same thing. In $\Bbb R$ is sequential compactness equivalent to compactness? I see some definitions of Heine–Borel theorem use compactness, and others use sequential compactness.

share|cite|improve this question

$\mathbb{R}$ and $\mathbb{R}^n$ are metric spaces (with metric given by $d(x,y)=\lvert x-y\rvert$).

share|cite|improve this answer
You probably mean $||x-y||$ in $\mathbb R^n$. – user18119 Oct 10 '12 at 20:20
@QiL Yes, sort of, though it is not at all uncommon to use the $\lvert\cdot\rvert$ notation even in $\mathbb{R}^n$. It depends on context. Neither notation is clearly wrong. – Harald Hanche-Olsen Oct 11 '12 at 8:08

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.