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Topology: Homeomorphic Mapping

  • Thread starter tylerc1991
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Homework Statement



Let X = [0,2 PI] x [0,2 PI] and let D be the partition of X that contains:
-{(x,y)} when x =/= 0,2 PI and y =/= 0,2 PI
-{(0,y),(2 PI,y)} for 0 <= y <= 2 PI
-{(x,0),(x,2 PI)} for 0 <= x <= 2 PI

Let U be the quotient topology on D induced by the natural map p: X -> D.

Prove that (D,U) is homeomorphic to S^1 x S^1 (let this = Y)

where S^1 = {(x,y) in R^2 : x^2 + y^2 = 1}

The Attempt at a Solution



OUTLINE:
Instead of going from D to Y I was thinking of going from X to Y. Essentially I would be going from a 2 PI by 2 PI square onto the torus of inner radius 0 and outer radius 2. Say this function is f. Then I could show that f is homeomorphic. Then I have a function p relating X and D, as well as a function f relating X and Y. Then I could relate D and Y and be done with the problem.

Is this the right idea for the problem? If so, I have been messing around with a few different functions for f but can't map that square onto the torus for the life of me. Can anyone give me a kind hand of assistance please?
 

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