15_4_14 - S , but cannot be shrunk through S to a point...

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14. a) S = { ( x , y ) : x > 0 , y 0 } is a simply connected domain. b) S = { ( x , y ) : x = 0 , y 0 } is not a domain. (It has empty interior.) c) S = { ( x , y ) : x 6= 0 , y > 0 } is a domain but is not connected. There is no path in S from ( - 1 , 1 ) to ( 1 , 1 ) . d) S = { ( x , y , z ) : x 2 > 1 } is a domain but is not connected. There is no path in S from ( - 2 , 0 , 0 ) to ( 2 , 0 , 0 ) . e) S = { ( x , y , z ) : x 2 + y 2 > 1 } is a connected domain but is not simply connected. The circle x 2 + y 2 = 2, z = 0 lies in
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Unformatted text preview: S , but cannot be shrunk through S to a point since it surrounds the cylinder x 2 + y 2 ≤ 1 which is outside S . f) S = { ( x , y , z ) : x 2 + y 2 + z 2 > 1 } is a simply connected domain even though it has a ball-shaped “hole” in it....
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This note was uploaded on 03/08/2012 for the course MATH 120 taught by Professor Onurfen during the Spring '12 term at Middle East Technical University.

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