3

In general topological equivalence of metric spaces does not imply strong equivalence. Is it true if the metric space is compact? More specifically: Let $d$ and $d'$ be topologically equivalent metrics on $X$. Let $(X,d)$ be a compact metric space. Are $d$ and $d'$ strongly equivalent?

Community
  • 1
Matias Heikkilä
  • 2,542

1 Answers1

5

Let $X=[0,1]$, $d(x,y)=|x-y|$ and $d'(x,y)=\sqrt{|x-y|}$. Since the open $r$-balls of one metric are the open $\sqrt r$ or $r²$ balls of the other, these metrics are weakly equivalent. But they are not strongly equivalent because $\frac {\sqrt r}r\to 0$ as $r\to 0$.

Hagen von Eitzen
  • 374,180