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Math Help - Equivalent Metrics

  1. #1
    FLT
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    Equivalent Metrics

    Hi, I'd be greatful for any help on the following problem; I have to show that two metrics \rho and \sigma are equivalent if and only if every open ball B_s^{\sigma}(x) contains an open ball B_r^{\rho}(x) and every open ball B_r^{\rho}(x) contains an open ball B_s^{\sigma}(x).

    Here is my proof thus far:

    If \rho and \sigma are equivalent, then we have that c\rho(x,y) \leq \sigma(x,y) \leq C\rho(x,y). Now, taking \sigma(x,y) \leq C\rho(x,y)...

    Here is where I get stuck - I want to say the following:

    ...for large enough s, we can say that \sigma(x,y) < C\rho(x,y) < s, and so we have that \sigma(x,y) < s is indeed our set B_s^{\sigma}(x). Taking C\rho(x,y) < s, and setting r = \frac{s}{C}, we also have our set B_r^{\rho}(x), and so we conclude that B_s^{\sigma}(x) \subset B_r^{\rho}(x).

    I am not allowed to use any arguments using cauchy sequences - only the equivalence inequality relation used above. I'd be so grateful for any help provided!
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  2. #2
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    Two metrics are said to be equivalent if they generate the same topology, and you know that a basis for a metric topology consists of the set of open balls. So, given B_r^\sigma (x), you know that this is an open set in both topologies generated by \sigma and \rho, thus B_r^\sigma (x) can be written as a union of balls B_{s_i}^\rho (x_i), where x_i \in B_r^\sigma (x). One of these must contain x, and consequently must contain a ball of radius s centered at x.

    Do you also need help with the other direction?
    Last edited by Nyrox; February 14th 2010 at 12:00 PM.
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  3. #3
    FLT
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    Quote Originally Posted by Nyrox View Post
    Two metrics are said to be equivalent if they generate the same topology, and you know that a basis for a metric topology consists of the set of open balls. So, given B_r^\sigma (x), you know that this is an open set in both topologies generated by \sigma and \rho, thus B_r^\sigma (x) can be written as a union of balls B_{s_i}^\rho (x_i), where x_i \in B_r^\sigma (x). One of these must contain x, and consequently must contain a ball of radius s centered at x.

    Do you also need help with the other direction?
    Thanks for your response. As I said previously, we are only allowed to use the definition of equivalence given by the inequality in the proof. I understand that there are Topological and Cauchy sequence treatments of the proof, but I'd like to see how this can be proved using only the inequality.
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    However the inequality above is NOT the definition of equivalent metrics. It is a sufficient condition, but not necessary. Consider for instance \mathbb R with the standard metric \rho and a second metric \sigma given by: \sigma (x,y)=\min \{1,\rho (x,y)\}. These two are equivalent since

    \begin{cases}B_{\frac{r}{r+1}}^\sigma (x) \subseteq B_r^\rho (x)\\B_r^\rho (x)\subseteq B_r^\sigma (x)\end{cases}.

    The inequality, though, is not satisfied.

    The notion of equivalent metrics is purely topological, so you must use topological arguments
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  5. #5
    FLT
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    Quote Originally Posted by Nyrox View Post
    However the inequality above is NOT the definition of equivalent metrics. It is a sufficient condition, but not necessary. Consider for instance \mathbb R with the standard metric \rho and a second metric \sigma given by: \sigma (x,y)=\min \{1,\rho (x,y)\}. These two are equivalent since

    \begin{cases}B_{\frac{r}{r+1}}^\sigma (x) \subseteq B_r^\rho (x)\\B_r^\rho (x)\subseteq B_r^\sigma (x)\end{cases}.

    The inequality, though, is not satisfied.

    The notion of equivalent metrics is purely topological, so you must use topological arguments
    Why not? I can find a C and a c such that C\rho(x,y) \leq \sigma(x,y) \leq c\rho(x,y). If \sigma(x,y) is the minimum of \rho(x,y) and 1, then clearly \sigma(x,y) \leq \rho(x,y) for any x,y. Likewise, for the upper bound, I can pick some C such that \sigma(x,y) > C\rho(x,y), with that C satisfying 0 < C < 1. Where's the flaw in my argument?
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  6. #6
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    Quote Originally Posted by FLT View Post
    Likewise, for the upper bound, I can pick some C such that \sigma(x,y) > C\rho(x,y), with that C satisfying 0 < C < 1.
    This is false! To see why, observe that

    \rho (0,10^n)=10^n and \sigma (0,10^n)=1 for all n\in \mathbb N

    thus, the constant C satisfies C\leq 10^{-n}, for all n\in \mathbb N, so C\leq 0, resulting in a contradiction.
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  7. #7
    FLT
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    Quote Originally Posted by Nyrox View Post
    This is false! To see why, observe that

    \rho (0,10^n)=10^n and \sigma (0,10^n)=1 for all n\in \mathbb N

    thus, the constant C satisfies C\leq 10^{-n}, for all n\in \mathbb N, so C\leq 0, resulting in a contradiction.

    No, for any n\in \mathbb N, C will satisfy 0<C<1! But this is besides the point; I can still pick such a C! It doesn't matter what C I pick, I can always pick a small enough one, ad infinitum.
    Last edited by FLT; February 15th 2010 at 08:00 AM. Reason: Error
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    C is a constant: it cannot depend on n. The inequality condition doesn't state that for each pair (x,y) there exist a C_{(x,y)} and a c_{(x,y)} such that

    C_{(x,y)}\rho(x,y)\leq\sigma(x,y)\leq c_{(x,y)}\rho(x,y).

    All metrics satisfy this! What the condition states is that, there exist some fixed C and c, such that, for all x,y

    C\rho(x,y)\leq\sigma(x,y)\leq c\rho(x,y)

    But there isn't any number C>0 such that C<10^{-n} for all n. For each n there exists a 0<C<10^{-n}, but not for all n at the same time.
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  9. #9
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    Quote Originally Posted by FLT View Post
    Hi, I'd be greatful for any help on the following problem; I have to show that two metrics \rho and \sigma are equivalent if and only if every open ball B_s^{\sigma}(x) contains an open ball B_r^{\rho}(x) and every open ball B_r^{\rho}(x) contains an open ball B_s^{\sigma}(x).

    Here is my proof thus far:

    If \rho and \sigma are equivalent, then we have that c\rho(x,y) \leq \sigma(x,y) \leq C\rho(x,y). Now, taking \sigma(x,y) \leq C\rho(x,y)...

    Here is where I get stuck - I want to say the following:

    ...for large enough s, we can say that \sigma(x,y) < C\rho(x,y) < s, and so we have that \sigma(x,y) < s is indeed our set B_s^{\sigma}(x). Taking C\rho(x,y) < s, and setting r = \frac{s}{C}, we also have our set B_r^{\rho}(x), and so we conclude that B_s^{\sigma}(x) \subset B_r^{\rho}(x).

    I am not allowed to use any arguments using cauchy sequences - only the equivalence inequality relation used above. I'd be so grateful for any help provided!
    I agree with Nyrox. What definition of "equivalent metrics" are you using. I also think that two metrics are equivalent if they generate the same topology. Otherwise, the only other natural definition would be that they are literally the same metric space.
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  10. #10
    FLT
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    Quote Originally Posted by Drexel28 View Post
    I agree with Nyrox. What definition of "equivalent metrics" are you using. I also think that two metrics are equivalent if they generate the same topology. Otherwise, the only other natural definition would be that they are literally the same metric space.
    OK, the definition I'm using is the following:

    Two metrics \rho and \sigma on the same space X are said to be equivalent if there exists c,C > 0 and independent of x,y such that

     <br />
c\rho(x,y) \leq \sigma(x,y) \leq C\rho(x,y)<br />
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  11. #11
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    Well, it's obviously legitimate to use that inequality as a definition, but in such case you cannot prove that every open ball in one metric containing an open ball in the other metric implies the equivalence of metrics At least not without some extra condition.
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