Prove that the quotient S1 x 0 1 is homeomorphic to the sph

Prove that the quotient S^1 x [0, 1]/ ~ is homeomorphic to the sphere S^2, where ((x,y), t) ~ ((x\', y\'), t\') if either t = t\' = 1, t = t\' = 0 or ((x, y), t) = ((x\',y\'), t\').

Solution

Case I: Let t = t\' = 1

Then ((x,y), 1) ~((x\', y\'),1)

by T(x,y) = (x\',y\')

Hence The quotient is homeomorphic if t = t\' =1

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Case II: If t = t\' =0

then ((x,y) , 0) ~((x\', y\'),0)

Hence by defining T(x,y) = (x\',y\')

Quotient S is homeomorphic to sphere.

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Case III: ((x,y), t)= ((x\',y\'),t\')

the mapping T(x,y,t) = (x\',y\',t\')

proved that S is homeomorphic.