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Math Help - Topology proof: smallest topology and base for topology

  1. #1
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    Topology proof: smallest topology and base for topology

    Thm: If Τ1 and Τ2 are topologies for X and Τ*=∩{ Τ | Τ is a topology for X such that Τ1 ⊆ Τ, Τ2 ⊆ Τ},
    then
    (1) Τ* is the smallest topology for X containing both Τ1 and Τ2; and
    (2) {(G1∩G2)| G1 ∈ Τ1, G2 ∈ Τ2} is a base for Τ*

    We know:
    B is a base for Τ iff
    (1) B⊆Τ and,
    (2) For every G∈Τ, if p∈G, then there is B∈B such that p∈B⊆G.
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  2. #2
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    Re: Topology proof: smallest topology and base for topology

    Quote Originally Posted by Kiefer View Post
    Thm: If Τ1 and Τ2 are topologies for X and Τ*=∩{ Τ | Τ is a topology for X such that Τ1 ⊆ Τ, Τ2 ⊆ Τ}, then
    (1) Τ* is the smallest topology for X containing both Τ1 and Τ2; and
    (2) {(G1∩G2)| G1 ∈ Τ1, G2 ∈ Τ2} is a base for Τ*
    As I read this \mathcal{T}^* is the intersection of all typologies on X which contain \mathcal{T}_1~\&~\mathcal{T}_2

    Exactly what difficulty are you having with part (1)? Please be complete in answering ?

    For part (2), note that if G\in\mathcal{T}_1 then G\in\mathcal{T}^*. WHY?

    Also G_1\in\mathcal{T}_1~\&~G_2\in\mathcal{T}_2 are in then G_1\cap G_2 must belong to \mathcal{T}^* because typologies are closed under finite intersection.
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  3. #3
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    Re: Topology proof: smallest topology and base for topology

    For part one, I can prove that T* is a topology but am not sure how to prove it is the smallest one in X containing T1 & T2
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    Re: Topology proof: smallest topology and base for topology

    Quote Originally Posted by Kiefer View Post
    For part one, I can prove that T* is a topology but am not sure how to prove it is the smallest one in X containing T1 & T2
    Why is that?
    Say \mathcal{T}_0 is smaller.
    But haven't you already incuded \mathcal{T}_0 in \mathcal{T}^*~?
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