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Math Help - question in metric topology

  1. #1
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    question in metric topology

    Let (X,d) be a metric space, Given a closed subset Y of X and a point x in X. Show that d(x,Y) is continuous as a function of x. that is f(x)=d(x,Y) is continuous

    Show also that a topology induced by the metric d is finer than the cofinite topology on X.

    Thank you very much for any help or guidance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by fuzzy topology View Post
    Let (X,d) be a metric space, Given a closed subset Y of X and a point x in X. Show that d(x,Y) is continuous as a function of x. that is f(x)=d(x,Y) is continuous

    Show also that a topology induced by the metric d is finer than the cofinite topology on X.

    Thank you very much for any help or guidance
    The first one is just messing around with the triangle inequality, what have you tried? For the second one let \mathcal{T}_C be the cofinite topology on X and \mathcal{T}_C the metric. Note then that if U\in\mathcal{T}_M then \#\left(X-U\right)<\infty and thus X-U is in \left\{X-O:O\in\mathcal{T}_M\right\} since any metric space is T_1 and thus U\in\mathcal{T}_M.


    (just for the record, the cofinite topology on a space is the coarsest T_1 topology)
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  3. #3
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    For the first question, I tried the following
    given epsilon > 0 , there is delta > 0, such that if d(x,y)<delta, then d(f(x),f(y)) should be less than epsilon, that is what we want.
    But d(x,y) < d(x,Y)+d(y,Y) =f(x)+f(y)

    Unfortunately I can not reach to the desired goal. Is what I wrote true. Please help me and thank you very very much for helping in solving the second question
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  4. #4
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    This is the theorem that you must prove.
    If A is a set and p~\&~q are points then \left| {D(A;p) - D(A;q)} \right| \leqslant d(p,q).
    Hints: Show that  \left( {\forall a \in A} \right)\left[ {\text{glb} \left\{ {d(x,a):x \in A} \right\} \leqslant d(p,a)} \right] then
    do it for q and observe D(A,p)-D(A,q)\le d(p,q).
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