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Math Help - Topology Boundry Question

  1. #1
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    Topology Boundry Question

    For each n\in Z_+, let B_n = {n, n+1, n+2, ....} and consider the collection B = {B_n | n\in Z_+}

    (a) Show that B is a basis for a topology for Z^+

    (b) Show thte the topology on X generated by B is not Hausdorff

    (c) Show that the sequence (2,4,6,8,...) converges to every point in Z_+ with the topology generated by B
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  2. #2
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    Quote Originally Posted by flaming View Post
    For each n\in Z_+, let B_n = {n, n+1, n+2, ....} and consider the collection B = {B_n | n\in Z_+}

    (a) Show that B is a basis for a topology for Z^+

    (b) Show thte the topology on X generated by B is not Hausdorff

    (c) Show that the sequence (2,4,6,8,...) converges to every point in Z_+ with the topology generated by B
    (a) The followings are the basis elements of B.

    B1={1,2,3,.................}
    B2={2,3,4,...............}
    B3={3,4,5,............}
    ........
    Bn={n, n+1, n+2,,,,}
    ...........

    To check B is a basis for Z+, you first need to check the union of the above elements are indeed Z+.
    Second, you need to verify that for each x in the interesection of the basis elements, there exists a basis element containing x and contained in the intersection.

    For instance, consider the intersection of B1 and B2. For each x in the section, you can find a basis element that containing x.

    (b) Pick two numbers, for instance, 3 and 5. Check open sets containing 3 & 5 respectively. Those open sets cannot be chosen disjointly.

    (c) Every open set containing a number in Z+ contains all but finite numbers of the above sequence in your topology. Thus, it converges everywhere in Z+.
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