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Thread: closed open both or neither sets in metric space

  1. #1
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    closed open both or neither sets in metric space

    let $d((a,b),(c,d))=d_{0}(a,c)+|b-d|$ be a metric on $\mathbb R^2$


    let
    X=$\{(x,0):x\in \mathbb R \}$ with the induced metric $d_{X}$


    Y=$\{(0,y):y\in \mathbb R \}$ with the induced metric $d_{Y}$


    A=$\{(x,y):x^2+y^2\leq4\}$


    B=$\{(x,y):x^2+y^2<9\}$




    i have to decide if the following subsets of X and Y are open,closed, neither or both.


    1) $A \cap X$ with respect to $d_{X}$


    2) $B \cap X$ with respect to $d_{X}$




    3) $A \cap Y$ with respect to $d_{Y}$


    4) $A \cap Y$ with respect to $d_{Y}$


    can someone check i have this correct.




    ive said that $d_{X}$, the restriction of d to X is just the discrete metric $d_{0}$ so i have both 1 and 2 are open and closed.


    and $d_{Y}$ is just the euclidean metric so 3 is closed and 4 is open
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  2. #2
    MHF Contributor
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    Re: closed open both or neither sets in metric space

    Hey jiboom.

    Have you tried setting a couple of variables to zero [which would happen in the dx and dy] and then testing that against the axioms? [Think of what happens when you take two constraints and combine them and set them consistent when the combination takes place].
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