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Math Help - Topology Proof

  1. #1
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    Topology Proof

    If T1 and T2 are topologies for X and B={(G1∩G2)|G1∈T1, G2∈T2},
    Then T*={G|For all p, if p∈G, then A∈B such that p∈A⊆G} is a topology.
    So I need to show
    (1) T*⊆P(X)
    (2) ∅∈T* and X∈T*
    (3) if A,B∈T*, then A∩B∈T*
    (4) if A⊆T*, then ∪A∈T*


    This is part of a problem proving T~=∩{T|T is a topology for X such that T1 ⊆ T and T2 ⊆ T}=>{(G1∩G2)|G1∈T1, G2∈T2} is a base for T*...So is G an element of T~ (not sure what else it would be an element of)?

    (I know the first two are kind of trivial, but if you are able to help me prove them anyways I would appreciate it)
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  2. #2
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    Re: Topology Proof

    Quote Originally Posted by Kiefer View Post
    If T1 and T2 are topologies for X and B={(G1∩G2)|G1∈T1, G2∈T2},
    Then T*={G|For all p, if p∈G, then A∈B such that p∈A⊆G} is a topology.
    So I need to show
    (1) T*⊆P(X)
    (2) ∅∈T* and X∈T*
    (3) if A,B∈T*, then A∩B∈T*
    (4) if A⊆T*, then ∪A∈T*
    I find your notation so hard to follow.
    (3) If A\in\mathcal{B}~\&~B\in\mathcal{B} then \exists G_1\in\mathcal{T}_1~\&~\exists G_2\in\mathcal{T}_2 such that A=G_1\cap G_2;
    \exists H_1\in\mathcal{T}_1~\&~\exists H_2\in\mathcal{T}_2 such that B=H_1\cap H_2.

    A\cap B=(G_1\cap G_2)\cap(H_1\cap H_2)=(G_1\cap H_1)\cap(G_2\cap H_2) but (G_1\cap H_1)\in\mathcal{T}_1~\&~(G_2\cap H_2)\in\mathcal{T}_2
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