solutionshw3-201141

solutionshw3-201141 - Solutions for HW 3 Problem 47 4 Sets...

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Solutions for HW 3 Problem 47: 4. Sets consisting of one point are obviously connected. Let E R \ Q contain at least two points a < b . Then there exist a rational number r with a < r < b . Let G 1 = ( -∞ ,r ) and G 2 = ( r, ). Then G 1 ,G 2 are disjoint open sets with union R \ { r } . Now a E G 1 and b E G 2 . Hence E G 1 6 = and E G 2 6 = , but E G 1 G 2 = and G 1 G 2 E . Hence E is not connected. Problem 47: 5. Let G 1 and G 2 be open sets such that G 1 E 6 = , G 2 E 6 = , G 1 G 2 E = , and G 1 G 2 E . Let x G 1 E . Then there exists a r > 0 such that B ( x ; r ) G 1 . Also B ( x ; r ) \ { x } ∩ E 6 = . Hence G 1 E 6 = . Similarly G 1 E 6 = . As E G 1 G 2 = and G 1 G 2 E are obvious, it follows that E is not connected. Contradiction. If E = (0 , 1) (1 , 2), then E = [0 , 2] is connected, but E is not. Problem 47: 6. Let
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This note was uploaded on 02/05/2012 for the course MATH 703 taught by Professor Schep during the Fall '11 term at South Carolina.

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