I'm trying to prove that a metric space is locally compact iff every closed ball is compact, using the more general definition that applies to Hausdorff spaces, that every point has a compact neighbourhood.
So call $X$ my space. The only non trivial thing to prove is that every closed ball is compact, assuming $X$ is locally compact. So consider $N$ a compact neighbourhood of some $x\in X$. Then as a neighbourhood, it contains $B(x,r)$ for some $r$. So it contains $\bar{B}(x,r/2)$. This is closed inside $N$ which is compact, so it's also compact. So I've proven that at any point there is a compact closed neighbourhood ball. Surely it's not too hard to prove all the bigger closed balls are compact ?
[Math] Locally compact metric space
compactnessmetric-spaces
Best Answer
Consider $R^2-\{(0,0)\}$ endowed with the canonical metric, it is locally compact. But $B((0,1);2)$ is not compact.
But the result is true if $X$ is endowed with a norm.