Prove that two metrics $d_1$ and $d_2$ on a set $X$ are equivalent if and only if there exists a metric space $(Y,\rho)$ and a homeomorphism $h:X\rightarrow Y$ from $X$ onto $Y$ such that $d_2(x,y)= \rho (h(x),h(y))$ for $x,y \in X$.

I know the basic definitions and properties of homeomorphism and equivalent metrics, but I am blank on this. Any idea on how to proceed?

  • $\begingroup$ There is something wrong with this question. The RHS of the equivalence does not mention $d_1$ and is always true. (Take $Y=X$, $\rho=d_2$.) The LHS is obviously not always true. $\endgroup$ – user491874 Jan 11 '18 at 10:53
  • $\begingroup$ @user8734617 Thanks for the hint. But, why is the LHS not always true? Do you mean that LHS and RHS are not equivalent? I came across this exercise in a text. $\endgroup$ – Kappa Jan 11 '18 at 11:02
  • $\begingroup$ Because not every two metrics are equivalent. (Take for example ordinary metric on $\mathbb R^n$ and discrete metric on the same set.) $\endgroup$ – user491874 Jan 11 '18 at 11:13
  • $\begingroup$ @user8734617 Of course. We have to assume that they are equivalent in order to show RHS is true. $\endgroup$ – Kappa Jan 11 '18 at 11:14
  • $\begingroup$ RHS is always true, no matter what $d_1$ is, as $d_1$ is not even mentioned in the RHS. LHS is not true if $d_1$ and $d_2$ are not equivalent. $RHS$ will be true in that case as you will take $Y=X, \rho=d_2, h=id_X$. $\endgroup$ – user491874 Jan 11 '18 at 11:15

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