Let $X$ be a non-empty set and $d_{1}$ and $d_{2}$ be metrics on $X$. The following statements are equivalent:
(a) $d_{1}$ and $d_{2}$ are equivalent.
(b) $f:X\to X$ given by $f(x) = x$ is continuous if considered from $(X,d_{1})$ to $(X,d_{2})$, and considered from $(X,d_{2})$ to $(X,d_{1})$.
My attempt
I have been able to prove the implication $(a)\Rightarrow(b)$ as follows.
Once $d_{1}\sim d_{2}$, it results that $\alpha d_{2}(x,y) \leq d_{1}(x,y) \leq \beta d_{2}(x,y)$ for each pair $(x,y)\in X^{2}$, where $\alpha > 0$ and $\beta > 0$. Indeed, for every $\varepsilon > 0$, there corresponds a $\delta_{\varepsilon} = \alpha\varepsilon$ such that for every $x\in X$ one has that
\begin{align*}
d_{1}(x,x_{0}) < \delta_{\varepsilon} \Rightarrow \alpha d_{2}(x,x_{0}) \leq d_{1}(x,x_{0}) < \alpha\varepsilon \Rightarrow d_{2}(x,x_{0}) < \varepsilon.
\end{align*}
which proves the function $f:(X,d_{1})\to(X,d_{2})$ is continuous.
Conversely, for each $\varepsilon > 0$, there corresponds a $\delta_{\varepsilon} = \varepsilon/\beta$ such that for every $x\in X$ one has that
\begin{align*}
d_{2}(x,x_{0}) < \delta_{\varepsilon} \Rightarrow \frac{1}{\beta}d_{1}(x,x_{0}) \leq d_{2}(x,x_{0}) < \frac{\varepsilon}{\beta} \Rightarrow d_{1}(x,x_{0}) < \varepsilon.
\end{align*}
which proves the function $f:(X,d_{2})\to (X,d_{1})$ is continuous.
The problem arises when I try to prove the implication $(b)\Rightarrow(a)$. I have tried to apply the definition of continuity, but it did not lead me anywhere.
Could anyone help me telling how to approach it?
Best Answer
It seems that you misunderstood the meaning of $d_1 \sim d_2$. You define it as follows:
This relation is known as strong equivalence of metrics. With this relation $(b) \Rightarrow (a)$ is not true. The correct definition of $d_1 \sim d_2$ is
This is nothing else than the explicit formulation of the continuity of $id : (X,d_1) \to (X,d_2)$ and $id : (X,d_2) \to (X,d_1)$.
Clearly strong equivalence implies "ordinary" (also called "topological") equivalence. As an example that the converse fails take any metric space $(X,d)$ with an unbounded metric (this means that there exist $x_n, y_n \in X$ such that $d(x_n,y_n) \ge n$). Define $d'(x,y) = \min(d(x,y),1)$. It is easy to check that $d'$ is a metric on $X$ which is toplogically equivalent to $d$ (simply take $\delta = \min(\epsilon, 1)$).
$d$ and $d'$ are not strongly equivalent: We have $d(x_n,y_n) \ge n$ and $d'(x_n,y_n) = 1$, thus $d(x_n,y_n) \ge n d'(x_n,y_n)$ which shows that no $\beta > 0$ can satisfy $d(x,y) < \beta d'(x,y)$ for all $x,y \in X$.