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Math Help - Define a meteric on a X so the associated topology is trivial

  1. #1
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    Define a meteric on a X so the associated topology is trivial

    What is the topology determined by the meteric on X given by d(x,y)=1 for x doesnt equal y and d(x,y)=0 for x=y?

    for me it looks the discrete topology. Is it right?

    can we define a meteric on a given set X so that the associated meteric topology is trivial.!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Define a meteric on a X so the associated topology is trivial

    Quote Originally Posted by rqeeb View Post
    for me it looks the discrete topology. Is it right?
    Right, for every x\in X we have B(x,1/2)=\{x\} which implies \{x\} is open etc.
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    Re: Define a meteric on a X so the associated topology is trivial

    Quote Originally Posted by FernandoRevilla View Post
    Right, for every x\in X we have B(x,1/2)=\{x\} which implies \{x\} is open etc.
    thank u sir
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    Re: Define a meteric on a X so the associated topology is trivial

    can any one help me in the 2nd part
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Define a meteric on a X so the associated topology is trivial

    Quote Originally Posted by rqeeb View Post
    can any one help me in the 2nd part
    Suppose X has at least two different elements x and y and denote d=d(x,y)>0. Then, B(x,d/2) is a nonempty open set and y\not\in B(x,d/2) . We can conclude.
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