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).