104_Fall09_MT2_sol

104_Fall09_MT2_sol - Midterm 2 Solutions for MATH 104,...

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Midterm 2 Solutions for MATH 104, Section 1 November 5, 2009 Show your work and justify your answers. 48 points total 1. Define the following: (a) (4 pts.) a metric space ; (b) (4 pts.) a disconnected metric space. 2. (4 pts.) State the inheritance principle. (see the book for answers to these problems) 3. (12 pts.) Consider a function f : M N , where M and N are metric spaces. Prove that if f is continuous then its graph is closed as a subset of M × N . Recall that the graph is graph ( f ) = { ( p,y ) M × N : y = f ( p ) } . Solution: Let F : M × N R be defined by F ( p,y ) = d ( f ( p ) ,y ). This is a composition of continuous functions (the distance function d is continuous by Ex- ercise 2.40), and therefore is continuous by Corollary 2.3. Also, if F ( p,y ) = 0, then d ( f ( p ) ,y ) = 0 which means that f ( p ) = y by the positive definiteness of the metric d . Then graph ( f ) = { ( p,y ) M × N : F ( p,y ) = 0 } = F - 1 [ { 0 } ]. Since { 0 } is a closed subset of R , F - 1 [ { 0 } ] is a closed subset of M × N by Theorem 10(ii), so graph ( f ) is closed. 4. Either prove that the statement is true, or prove it is false. (a) (6 pts.) There exists a continuous surjective map from the circle
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104_Fall09_MT2_sol - Midterm 2 Solutions for MATH 104,...

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