Give example of a set $X$ and two metrics $d_1,d_2$ on $X$ such that $(X,d_1)$ and $(X,d_2)$ are topologically equivalent but there exist a function $f:X \to X$ which is uniformly $d_1$ continuous but not uniformly $d_2$ continuous . What I know is that the metrics cannot be strongly equivalent . Please help . Thanks in advance

  • $\begingroup$ I think it's just "metrics" and not "metrices" ;) $\endgroup$ – Math1000 Feb 28 '15 at 14:26
  • $\begingroup$ @Math1000 : I will edit :) $\endgroup$ – user217921 Feb 28 '15 at 14:28

Let $X=(0,1)$, and let $d_1$ be the usual Euclidean metric on $X$. For $x,y\in X$ let $$d_2(x,y)=\left|\frac1x-\frac1y\right|\;.$$

  • Verify that $d_2$ is a metric on $X$ and is topologically equivalent to $d_2$.
  • Consider the function $f(x)=1-x$.

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