[Math] Example of a metric space where Heine-Borel theorem does not hold

Question

In Rudin, it says the Heine-Borel theorem holds for Euclidean metric spaces.

What is an example of a metric space where Heine-Borel does not hold true?

Answer 1

Consider $\Bbb R^2 \setminus \{(0,0)\}$ with the usual metric restricted from $\Bbb R^2$. The set $$D = \{ (x,y) \in \Bbb R^2 \mid 0 < x^2+y^2 \leq 1 \}$$is closed in $\Bbb R^2 \setminus \{(0,0)\}$, bounded, but not compact. Sequences in $D$ which "want" to converge to $(0,0)$ don't have limit in $D$.