final-answers - 7510 Fall 06 Final Answers B1. Let L be a...

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Unformatted text preview: 7510 Fall 06 Final Answers B1. Let L be a straight line in the plane. The topology that L inherits as a subspace of R lscript R lscript is a familiar topology; what is it? (The answer will be different for different lines.) Justify your answer. Solution. Consider a basis element B = [ a 1 ,b 1 ) [ a 2 ,b 2 ) for R 2 lscript . Sup- pose that L has negative slope. Then L B = { a 1 a 2 } if a 1 a 2 L , so every singleton is open in L and L has the discrete topology. Oth- erwise L B is either empty or is sent to an interval [ a,b ) under some fixed (linear) bijection L R , so L is homeomorphic to R lscript . square B2. Define an equivalence relation on R 2 by x 1 y 1 x 2 y 2 if x 2 1 + y 2 1 = x 2 2 + y 2 2 . The quotient space R 2 / is a familiar space; what is it? Justify your answer. Solution. Let p : R 2 R 2 / be the quotient map. Define f : R 2 [0 , ) by f ( x y ) = x 2 + y 2 . Then f is a continuous surjection, and f ( x 1 y 1 ) = f ( x 2 y 2 ) iff x 1...
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final-answers - 7510 Fall 06 Final Answers B1. Let L be a...

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