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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!

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    Do you mean that those inequalities are supposed to hold for *all* elements $x$ and $y$ of $X$, or what?2012-11-24
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    what i understand from the question is, not for all x and y, these are just two random metrics. Am i right?2012-11-24
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    $x$ and $y$ are certainly not random metrics. You said yourself they are elements of $X$.2012-11-24

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