I got stuck on this problem. Hope someone can give some hint to move on. Thanks.
Suppose $d_1(x,y) = |x-y|$, $d_2(x,y)=|\phi(x) – \phi(y)|$ where $\phi(x) = {x \over {1 + |x|}}$. Prove that $d_1$ and $d_2$ are metrics on $\mathbb{R}$ which induce the same topology, although $d_1$ is complete and $d_2$ is not.
For this problem, I can prove the part $d_1$ and $d_2$ are both metrics, $d_1$ is complete and $d_2$ is not. But I got stuck on the part to prove that two metrics induce the same topology (meaning that if $U$ is open in $(X,d_1)$, then it must be open in $(X, d_2)$).
More detail, for example, given $\epsilon > 0$, I need to find $\delta > 0$ such that if $d_2(x,y) < \delta$, then $d_1(x,y) < \epsilon$ and vice versa. Can anyone give me some hint to find those $\delta$. Thanks
Best Answer
The last statement is false: if you could always find such $\delta$ for each $\varepsilon$ such that it held for any pair $x,y$, and vice versa, the metrics would be uniformly equivalent. This implies that as uniform spaces they have the same uniformity and so their notion of completeness would also agree.
What you need to prove is that each ball with respect to $d_1$ contains one (with the same centre) with respect to $d_2$ and vice versa.
So you need
$$\forall x \in X \forall \varepsilon > 0 \exists \delta > 0 : B_{d_2}(x,\delta) \subseteq B_{d_1}(x,\varepsilon)\text{,}$$ and vice versa with $d_1$ and $d_2$ interchanged.
The inclusion can also be written as a quantifier statement and then gives:
$$\forall x \in X \forall \varepsilon > 0 \exists \delta > 0 : \forall y \in X: ( d_2(x,y) < \delta \rightarrow d_1(x,y) < \varepsilon )$$
(And the reverse too.) Note that the $\delta$ can depend on $x$ and in your statement it does not: you choose the same $\delta$ for all $x,y$.
As $d_2(x,y) = d(\phi(x),\phi(y))$ to prove this is closely related to showing $\phi$ continuous...