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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This note was uploaded on 01/16/2010 for the course MATH mat taught by Professor Mankres during the Spring '09 term at University of Alaska Southeast.

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

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