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Class 9 Homework-

Let X:={(u,v)R2:πuπ,1v1} (Rectangle) and let Y:={(x,y,z)R3:x2+y2=1,|z|1} (Cylinder). Verify the following:

(1) If uu=±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:XY,f(u,v)=(cosu,sinu,v) is a continuous surjection. Conclude that X/ and Y are homeomorphic.




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