Posted by J. Holmer on October 8, 2007, 4:41 pm, in reply to "Problem 4d" In this problem, we are taking two disjoint copies of the closed unit disc and then gluing them on the boundary. So first we form the disjiont union and on the disjoint union we assign the following topology: a set U in the disjoint union is open iff U\cap D_1 and U\cap D_2 are both open. The next step is to define the equivalence relation given and put the quotient topology on the set of equivalence classes (the quotient set). If you prefer, you can set up the problem like this. Let D_1 = {(x,y,0)\in R^3 | x^2+y^2 \leq 1 } and D_2 = {(x,y,1)\in R^3 | x^2+y^2 \leq 1 } Thus we are putting D_1 in the z=0 plane of R^3 and D_2 in the z=1 plane of R^3. Then we declare that (x_1,y_1,0) \sim (x_2,y_2,1) iff (x_1,y_1)=(x_2,y_2).
Board Administrator
Let me clarify: Two points (x_1,y_1), (x_2,y_2) in the borders of D_1,D_2 respectively are declared to be equivalent iff (x_1,y_1)=(x_2,y_2).
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