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

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Unformatted text preview: Math W4051: Problem Set 4 due Wednesday, October 3 Reading: Munkres, Ch.3. 1. Define an equivalence relation on the space X = R2 - {0} as follows: (x1 , y1 ) (x2 , y2 ) if and only if there exists k Z such that (x1 , y1 ) = (2k x2 , 2k y2 ). The space X R2 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 f is a homeomorphism. Correct formulas will suffice.) 2. Munkres, Ch.2 22, exercise 4 on p. 145. (Again, it suffices to write down correct formulas for the homeomorphisms and their inverses.) 3. Munkres, Ch.3 23, exercises 5, 7, 8 on p. 152. 4. Munkres, Ch.3 24, exercises 1(a), 1(c), 2 on p. 157-158. 5. Munkres, Ch.3 25, exercise 5 on p. 162. ...
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This note was uploaded on 02/10/2010 for the course MATH 205 taught by Professor Smith during the Spring '05 term at Adrian College.

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