Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
2  
What have you tried? –  Chris Eagle Nov 24 '12 at 14:22
    
Do you mean that those inequalities are supposed to hold for all elements $x$ and $y$ of $X$, or what? –  Chris Eagle Nov 24 '12 at 14:24
    
what i understand from the question is, not for all x and y, these are just two random metrics. Am i right? –  user49065 Nov 24 '12 at 14:25
    
$x$ and $y$ are certainly not random metrics. You said yourself they are elements of $X$. –  Chris Eagle Nov 24 '12 at 14:26
add comment

2 Answers 2

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.

share|improve this answer
add comment

Let $\tau_j$ be the topology induced by $d_j$ for $j=1,2$, and assume $B(x;r)$ be an open ball in $\tau_1$. It suffices to show there is an open ball $B(x;\delta)$ in $\tau_2$ such that $B(x;\delta)\subset B(x;r)$. (In this way we show that every open set in $\tau_1$ is also open in $\tau_2$.)

But since the two metrics are equivalent, this is not difficult to do this once you pick $\delta$ small enough.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.