Openness w.r.t. these two metrics are equivalent. Suppose $(X,d)$ is a metric space. Define $\delta:X\times X\rightarrow[0,\infty)$, as $$\delta(x,y)=\frac{d(x,y)}{1+d(x,y)}.$$ It is easy to show that $\delta$ is a metric as well, but I am having difficulty in showing that if a subset of $X$ is $d$-open , then it is $\delta$-open too.
Thanks in advance!

Hint: $f\colon [0,\infty)\to[0,\infty), x\mapsto \frac{x}{1+x}$ is strictly increasing.
