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Thread: metric continuous function

  1. #1
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    metric continuous function

    Suppose (\mathbb{X},d) is a metric space and the corresponding product topology is defined on \mathbb{X}\times\mathbb{X}. Prove that the metric d is continuous.
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    Quote Originally Posted by raimis View Post
    Suppose (\mathbb{X},d) is a metric space and the corresponding product topology is defined on \mathbb{X}\times\mathbb{X}. Prove that the metric d is continuous.
    You have to show that if (x,y) is close to (x_0,y_0) in \mathbb{X}\times\mathbb{X}, then d(x,y) is close to d(x_0,y_0). More precisely, you have to show that, given (x_0,y_0)\in\mathbb{X}\times\mathbb{X} and \varepsilon>0, there exists \delta>0 such that if d(x,x_0)<\delta and d(y,y_0)<\delta then |d(x,y) - d(x_0,y_0)|<\varepsilon. Use the triangle inequality.
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    product topology

    And how do we make use of the product topology here?
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    Quote Originally Posted by raimis View Post
    And how do we make use of the product topology here?
    The condition d(x,x_0)<\delta and d(y,y_0)<\delta expresses the fact that (x,y) is close to (x_0,y_0) in the product topology.
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    Quote Originally Posted by raimis View Post
    Suppose (\mathbb{X},d) is a metric space and the corresponding product topology is defined on \mathbb{X}\times\mathbb{X}. Prove that the metric d is continuous.
    I'm sorry. What is this question actually asking. I see that Opalg has already answered it, but I am just curious. Is it that if (X,d) is a metric space then d:X^2\mapsto\mathbb{R} is continuous?
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    Quote Originally Posted by Drexel28 View Post
    Is it that if (X,d) is a metric space then d:X^2\mapsto\mathbb{R} is continuous?
    Yes, the question is as you say. But the problem involves product topology and I am wondering whether it should necessarily be defined on \mathbb{X}\times\mathbb{X} to prove the statement.
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