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Math Help - Can open balls and pairwise intersections of balls generate a topology?

  1. #1
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    Can open balls and pairwise intersections of balls generate a topology?

    Hi,

    I have a question. Can the following topology base generate a topology(a family of sets)?:

    Can open balls and pairwise intersections of balls generate a topology?-baza.png


    X - finite set of points, d - metric
    B - open ball in X
    r - ray, const value

    Additionally, how could be the cover of a family of sets in topology realized ?

    Thans in advance,
    algorytmus
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  2. #2
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    Re: Can open balls and pairwise intersections of balls generate a topology?

    A family \mathcal{F} of sets generates a topology on X just in case:

    (1) \mathcal{F}\subseteq 2^X;
    (2) x\in X \implies \exists B\in\mathcal{F} such that x\in B;
    (3) x\in B_1\cap B_2 for B_1,B_2\in\mathcal{F} \implies \exists B_3\in\mathcal{F} such that x\in B_3\subseteq B_1\cap B_2.

    Your collection satisfies these requirements. In fact, notice that the second term of the union is superfluous, and that \mathcal{B}=\{B_d(x,r):x\in X,r>0\}. Given that X is finite, this means \mathcal{B}=2^X\setminus\emptyset, i.e. it generates the discrete topology.

    I'm not sure what you mean by realizing a cover of a family of sets though.
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  3. #3
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    Re: Can open balls and pairwise intersections of balls generate a topology?

    Thank you for help!

    Could you correct me if I am wrong in the following:

    Let me illustrate examples geometrically.

    Given X={a, b, c}


    the topological base is

    \mathcal{B}={{B_a},{B_b},{B_c},{B_a,B_b},{B_b,B_c}}

    ,
    for


    it is:

    \mathcal{B}={{B_a},{B_b},{B_c}}

    B_x - a ball for x
    Last edited by algorytmus; October 7th 2011 at 04:43 PM.
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  4. #4
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    Re: Can open balls and pairwise intersections of balls generate a topology?

    Well if you mean that \mathcal{B}=\{\{x\}:x\in X\}, then it still generates the discrete topology.
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