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Math Help - Proving a set open

  1. #1
    Member thaopanda's Avatar
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    Proving a set open

    I need help with this proof...

    d_{1},d_{2} : X \times X \rightarrow R distance on X are called equivalent provided that there exists C_{1},C_{2} > 0 such that
    C_{2}d_{2}(x,y) \leq d_{1}(x,y) \leq C_{1}d_{2}(x,y)
    \forall x,y \epsilon X

    Let d_{1},d_{2} be equivalent distances on X and let A \subseteq X. Prove that A is open in (X,d_{1}) if and only if A is open in (X,d_{2}) (i.e. equivalent distances give the same open sets).

    please help
    Nicole
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  2. #2
    Senior Member
    Joined
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    Quote Originally Posted by thaopanda View Post
    I need help with this proof...

    d_{1},d_{2} : X \times X \rightarrow R distance on X are called equivalent provided that there exists C_{1},C_{2} > 0 such that
    C_{2}d_{2}(x,y) \leq d_{1}(x,y) \leq C_{1}d_{2}(x,y)
    \forall x,y \epsilon X

    Let d_{1},d_{2} be equivalent distances on X and let A \subseteq X. Prove that A is open in (X,d_{1}) if and only if A is open in (X,d_{2}) (i.e. equivalent distancesgive the same open sets).

    please help
    Nicole
    Lemma 1. Let d_1, d_2 be two metrics for the set X. If there exists a positive number c_1 such that d_{1}(x,y) \leq c_{1}d_{2}(x,y) for all x,y \in X. Then the identity map X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous.

    Proof. Let p \in X and let \epsilon be a positive number. If \delta = \frac{\epsilon}{c_1} and x is a member of X for which d_2(x,p) < \delta, then d_1(i(x), i(p)) = d_1(x, p) \leq c_1d_2(x,p) < c_1\delta = \epsilon. This shows that d_1(i(x), i(p)) < \epsilon whenever d_2(x,p) < \delta . Thus X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous.

    I'll leave it to you to show that X, d_1) \rightarrow (X, d_2)" alt="i^{-1}X, d_1) \rightarrow (X, d_2)" /> is continuous.

    Since X, d_2) \rightarrow (X, d_1)" alt="iX, d_2) \rightarrow (X, d_1)" /> is continuous, then for each d_1-open set, i^{-1}(O) = O is d_2-open set. This argument applies for X, d_1) \rightarrow (X, d_2)" alt="i^{-1}X, d_1) \rightarrow (X, d_2)" /> as well. Thus, d_1, d_2 determines the same open sets in X.
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