Homework Help: Show that metrics d_1,d_2 are equivalent

1. Mar 19, 2012

Ratpigeon

1. The problem statement, all variables and given/known data

Show that two metrics d_1,d_2 are equivalent iff for all epsilon>0, exists delta>0 such that
B_(d_1)(x,epsilon) $\subset$ B_(d_2)(x,delta) and vice versa (Where B_(d_1)(x, epsilon) is the open ball on the metric d_1 around x with radius epsilon.

2. Relevant equations

pretty much what is in the first part

3. The attempt at a solution
This is what I tried:
Let B_(d_1)(x,epsilon)=I, which is an open interval;

Assume
a $\in$ B_(d_1)(x,epsilon) and
a not $\in$ B_(d_2)(x,deltamax)
I'm trying to show this leads to a contradiction of deltamax being the largest delta that gives B_(d_2)(x,delta)$\subset$ I, but I'm not sure how.

2. Mar 19, 2012

sunjin09

For the if part, all you have to show is an open set under d1 is also an open set under d2. Considering definition of open set in a metric space, i.e., each point of the set has an open ball contained in the set, this is almost obvious. The only if part is even more obvious following definition.

3. Mar 19, 2012

Ratpigeon

I have to show - insofar as I understand - that if they are equivalent metrics, then for any open ball for d1, there is an identical open ball for d2 ( because B_d1(x, epsilon) subset of B_d2(x,delta) and b_d2(x,delta) subset of B_d1(x,epsilon), which is more restrictive than the solution you suggested.
But thank you for your help.

4. Mar 19, 2012

Ratpigeon

I have to show - insofar as I understand - that if they are equivalent metrics, then for any open ball for d1, there is an identical open ball for d2 ( because B_d1(x, epsilon) subset of B_d2(x,delta) and b_d2(x,delta) subset of B_d1(x,epsilon), which is more restrictive than the solution you suggested.
But thank you for your help.

5. Mar 20, 2012

sunjin09

I see. Apparently I didn't know what equivalence of metric means. I was thinking of the equivalence between the two topologies.

6. Mar 20, 2012

Ratpigeon

The lecturer explained the problem after class - the notation had been unclear, and I'd misunderstood the problem. I've got it out now. But thanks - your suggestion was actually the right answer...

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