Let X:={(u,v)∈R2:−π≤u≤π,−1≤v≤1} (Rectangle) and let Y:={(x,y,z)∈R3:x2+y2=1,|z|≤1} (Cylinder). Verify the following:
(1) If u−u′=±2πthen define (u,v)∼(u′,v′) iff v=v′ . Otherwise, define (u,v)∼(u′,v′) iff (u,v)=(u′,v′). Then ∼ defines an equivalence relation on X.
(2) f:X→Y,f(u,v)=(cosu,sinu,v) is a continuous surjection. Conclude that X/∼ and Y are homeomorphic.