a1f08s - HINTS TO ASSIGNMENT 1(Since it is your first...

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HINTS TO ASSIGNMENT 1 ( Since it is your first assignment mostly from the part of mathematics that you are not familiar with, the hints are exceptionally generous, often just a solution to the problem ) 1. Consider the set 2 } 2 0 , 0 : ) , {( } 0 , 1 0 : ) , {( R x y y x x x y y x S = < = Then S x x y y x S = } 0 , 1 0 : ) , {( int . 2. NO. Let U be the union of two open rectangles } 1 0 , 1 0 : ) , {( 1 < < < < = y x y x U and . Since U is open, . The closure of U } 1 0 , 2 1 : ) , {( 2 < < < < = y x y x U U U = int is } 1 0 , 2 0 : ) , {( = y x y x U . Note that the point is in the interior of ) 2 / 1 , 1 ( U but is not in . int U 3. To prove that S is open, show that 0 , ) , ( > = r S y x x such that . S r B ) , ( x Choosing , show that if 1 = y r ) , ( ) , ( 1 1 x r B y x , then r y y < 1 and conclude that , that is and hence , what implies that S is open. 1
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