Are Two Metric Spaces Equivalent?

Question

Are the following metrics equivalent on $\Bbb R$? $ d(x,y)=|x-y|$ and $d'(x, y)=|\tan^{-1}(x) - \tan^{-1}(y)|$.

Answer 1

If the basic set is $\mathbb R$, yes they are equivalent. What you have to show is that if $x_n \to x$ in one metric then $x_n \to x$ in the other. This is obvious because $\tan$ and $\arctan$ are continuous functions in the usual topology.

Answer 2

Yes. They are equivalent. \In general if $f$ is any continuous increasing bijective function then the metrics defined by $$d(x,y)=|x-y|$$ and $$d'(x,y)=|f(x)-f(y)|$$ are equivalent.