Let \(X:=\{(u,v) \in \mathbb{R^2}:- \pi \le u \le \pi ,-1 \le v \le1\}\) (Rectangle) and let \(Y:=\{(x,y,z) \in \mathbb{R^3}:x^2+y^2=1,|z| \le1\}\) (Cylinder). Verify the following:
(1) If \(u-u^{\prime} =\pm2 \pi\)then define \((u,v) \sim(u^\prime ,v^\prime)\) iff \(v=v^\prime\) . Otherwise, define \((u,v) \sim(u^\prime ,v^\prime)\) iff \((u,v) = (u^\prime ,v^\prime)\). Then \(\sim\) defines an equivalence relation on \(X\).
(2) \(f:X \rightarrow Y,f(u,v)=(cosu,sinu,v)\) is a continuous surjection. Conclude that \(X / \sim\) and \(Y\) are homeomorphic.
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