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Masters Comprehensive Exam - Topology January 5, 2008 CHOOSE THREE (1) Suppose that f : X → Y is continuous and Y is Hausdorff. Prove that {(x1 , x2 ) : f (x1 ) = f (x2 ) } is a closed subset of X × X (with the product topology). (2) Suppose that f : X → Y is a continuous surjection. Prove: a: If X is compact then Y is compact. b: If X is connected then Y is connected. (3) Let X be the subspace of R3 , X = {(x, y, z) : x2 + y2 ≤ 1, z = −1} ∪ {(x, y, z) : x2 + y2 ≤ 1, z = 1}, and let ∼ be the equivalence relation defined on X by (x1 , y1 , z1 ) ∼ (x2 , y2 , z2 ) if and only if (x1 , y1 , z1 ) = (x2 , y2 , z2 ) or x1 = x2 , y1 = y2 , and x21 + y21 = 1. Prove that the quotient space X/ ∼ is homeomorphic with the sphere S2 = {(x, y, z) : x2 + y2 + z2 = 1}. (4) Suppose that γ : [0, 1] → X is a path from x0 to x1 . Define Γ : Ω(X, x0 ) → Ω(X, x1 ) (Ω(X, xi ) is the set of loops in X based at xi ) by γ(1 − 3t), 0 ≤ t ≤ 1/3, α(3t − 1), 1/3 ≤ t ≤ 2/3, Γ(α)(t) = γ(3t − 2), 2/3 ≤ t ≤ 1. Prove that: a: If α is path homotopic to β then Γ(α) is path homotopic to Γ(β). b: Γ(α ? β) is path homotopic to Γ(α) ? Γ(β).