<s>[INST] Proof verification: $d$ and $\tilde{d}$ are topologically equivalent Given $X = (0,1]$ a metric space with $\tilde{d}$ defined as $$\tilde{d}(x,y) = \bigl|\frac{1}{x} - \frac{1}{y}\bigr| \ \  \text{for} \ \ x,y \in X$$
I'm trying to prove that  $\tilde{d}$ and $d$ the standard metric on $(0,1]$ are topologically equivalent, using the open sets criteria and would like to know if it's correct.
Proof:
$U$ is open in $(X,\tilde{d}) \implies U$ is open in $(X,d)$
Let $x \in U$ and $\epsilon > 0$ such that $B_{\epsilon}(x,\tilde{d}) \subseteq U$
If $y \in B_{\epsilon}(x,\tilde{d})$ which means $ \tilde{d}(x,y) =\bigl|\frac{1}{x} - \frac{1}{y}\bigr| = \bigl|\frac{y-x}{xy}\bigr| = \frac{|x-y|}{|xy|} < \epsilon$ and that's equivalent to $d(x,y) = |x-y| < |xy|\epsilon$, then $y \in U$
Therefore, for  any $x \in U$ we can find an epsilon $\tilde{\epsilon} = |xy|\epsilon$ such that: $y \in B_{|xy|\epsilon}(x, d) \implies y \in U$
$\Longleftarrow$
Similarly, If $y \in B_{\epsilon}(x,d)$, it follows that $d(x,y) = |x-y| < \epsilon$ which is equivalent to $\tilde{d}(x,y) =\bigl|\frac{1}{x} - \frac{1}{y}\bigr| < \frac{\epsilon}{|xy|}$
Therefore, for  any $x \in U$ we can find an epsilon $\tilde{\epsilon} = \frac{\epsilon}{|xy|}$ such that: $y \in B_{\frac{\epsilon}{|xy|}}(x, \tilde{d}) \implies y \in U$
 [/INST] There is a fault in your proof. Kindly see the proof below for the other side
\begin{align}\bar{d}(x,y)<r_2 &\iff \left| \dfrac{1}{x}-\dfrac{1}{y}  \right| <r_2 \iff \dfrac{1}{x}-r_2<\dfrac{1}{y}<\dfrac{1}{x}+r_2\\&
\iff \dfrac{1-x r_2 }{x}<\dfrac{1}{y}<\dfrac{1+xr_2 }{x}
 \end{align} 
Choose $r_2<\dfrac{1}{x},$ then
\begin{align}
 \dfrac{1-x r_2 }{x}<\dfrac{1}{y}<\dfrac{1+xr_2 }{x}&\iff  \dfrac{x }{1+x r_2}<y<\dfrac{x }{1-xr_2}\\& \iff - \dfrac{x^2 r_2 }{1+x r_2}<y-x<\dfrac{x^2 r_2 }{1-xr_2}.
 \end{align} 
As $r_2\to 0,$ then $\dfrac{x^2 r_2 }{1+x r_2}\to 0$ and $\dfrac{x^2 r_2 }{1-xr_2}\to 0.$ Hence, $\forall \, r>0,$ and $\forall \,x\in (0,1],$ there exists $r_2\in (0,1]$ such that $r_2<(r/2x^2).$ So, 
\begin{align}
 y-x<\dfrac{x^2 r_2 }{1-xr_2}<x^2 r_2 <\dfrac{r  }{2}<r.
 \end{align} 
Also, \begin{align}
r_2<\dfrac{r}{2x^2}\iff -r_2>-\dfrac{r}{2x^2}\iff -x^2 r_2>-\dfrac{r}{2}.
 \end{align}
But, $r_2\in (0,1]\implies r_2>-\dfrac{1}{2x},$ implies $x r_2>-\dfrac{1}{2} \iff 1+x r_2>\dfrac{1}{2}\iff \dfrac{1}{1+x r_2}<2.$ Hence,
\begin{align}
y-x> - \dfrac{x^2 r_2 }{1+x r_2}>-\dfrac{r}{2(1+x r_2)} > -\dfrac{2r}{2}=-r
 \end{align} 
Hence, we have that 
\begin{align}
-r=-\dfrac{2r}{2}<-\dfrac{r}{2(1+x r_2)}< - \dfrac{x^2 r_2 }{1+x r_2}<y-x<\dfrac{x^2 r_2 }{1-xr_2}<x^2 r_2 <\dfrac{r  }{2}<r.
 \end{align} 
Therefore, \begin{align}
|x-y|<r \iff d(x,y)<r
 \end{align}
And you are done. Kindly get back if you have some questions. Remember that they don't necessarily need to have the same radius.
</s>