Poincare Conjecture Proof - Gauge-institute.org

Poincare Conjecture ProofH. Vic DannonProof: T is the smallest topology onTθ. Such topology is called sup topology [11].3Σ which is larger than eachIn [11], each of the topologies is defined on the same space X , and2 3contains it. Here, for each θ , Tθ⊆Σθ ⊆Eθ, and all the Tθaredisjoint from each other. { : ∈ [0,2 ]}∩T θ T θ=∅ for 1 21 2θ≠θTθθ π partitions T since, andT = ∪ Tθ.θ∈[0,2 π]5.63T is Sup Topology on S , and f : [0,2 ] partitionsθT θθ ∈ πT{ }Proof:fθ Tθ∩fθ Tθ=∅for θ 1≠ θ 2, and T = ∪ θ]f T θ.[0,21 1 2 2θ∈π25.7 For each θ , let Sθbe a Subbase for Tθon Σθ, with sets H θ.Then,∪S is a subbase for T , with subbase setsθθ∈[0,2 π]θ∈Θ∈τProof: If G ∈ T , there is∪Θ∈ τ , and there are G θ∈ T θ, so thatGFor each G θ, there are base sets= ∪ G θ.θ∈ΘjB θ, j∈ J , so thatHθGθ= ∪ Bj∈JjθFor eachjjk ,B θ, there are subbase sets H θ∈ S , k = 1... n so thatθ16