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hw4 - Math W4051 Problem Set 4 due Wednesday October 3...

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Math W4051: Problem Set 4 due Wednesday, October 3 Reading: Munkres, Ch.3. 1. Define an equivalence relation on the space X = R 2 - { 0 } as follows: ( x 1 , y 1 ) ( x 2 , y 2 ) if and only if there exists k Z such that ( x 1 , y 1 ) = (2 k x 2 , 2 k y 2 ) . The space X R 2 is given the standard (subspace) topology, and X * = X/ the corresponding quotient topology. Write down a formula for a homeomorphism f : X * T 2 and its inverse f - 1 : T 2 X * , where T 2 = [0 , 1] × [0 , 1] / (0 , x ) (1 , x ) , ( x, 0) ( x, 1) for all x [0 , 1] is the torus. (You do not need to justify in detail that
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