metrics, open sets i am studying for my exam and trying to solve some questions. I have got a problem about proving the following.
Let $X$ be a set, and let $d_1$ and $d_2$ be two metrics on $X$. Suppose that $d_1$ and $d_2$ are equivalent in the sense that there is a constant $C \ge 1$ such that $d_1(x,y) \le Cd_2(x,y)$; $d_2(x, y)\le Cd_1(x, y)$; and $x, y$ are elements of $X$. Show that the metric spaces $(X,d_1)$ and $(X,d_2)$ have the same open sets.
I would appreciate if someone can help. Thanks!

You have to prove that every open set for $d_1$ is an open set for $d_2$ and conversely (however here you do not even need to prove the converse considering the symmetry in the definition). So take an open set $O$ for $d_1$, and prove that it is an open set for $d_2$. You probably have a definition of open sets in a metric space that says "A subset $O$ of $X$ is open iff for every $x \in O$, there is an open ball centered in $x$ within $O$". The rest should be easy.
