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week25 - Topology(Part 1 Original Notes adopted from April...

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Topology (Part 1) Original Notes adopted from April 2, 2002 (Week 25) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong R, A subset S of R is open if S is a union of open intervals; ie) whenever x S , there exists open interval (a,b) such that x (a,b) & (a,b) S. 7 -2.1 3 7 9 13 17 S = { x: x< -7 or x (-2.1, 3) or x (7,9) or x (13,17)} In R 2 , an open disk is a set of form {(x,y): (x-a) 2 + (y-b) 2 < r} ie) an integer of circle. R 2 , A subset S of R 2 is open if S is a union of open disks; ie) whenever (x,y) S , there exists open disk D such that x D & D S. eg. {(x,y): y > x 2 } Note: (in R and R 2 ): the union of any number of open sets is open. Eg. Let In = (-1/n, 1/n) In = {0} not open. n=1 The intersection of open sets need not be open. The intersection of 2 open sets is open. Let S 1 ,S 2 be open and suppose x S 1 S 2 there exists, (a 1 ,b 1 ) such that x (a 1 ,b 1 ) & (a 1 ,b 1 ) S 1 there exists (a 2 ,b 2 ) such that x (a 2 ,b 2 ) & (a 2 ,b 2 ) S 2 a 1 a 2 x b 1 b 2 (a 2 ,b 1 ) S 1 S 2 (S 1 S 2 ) S 3 We define to be open.
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week25 - Topology(Part 1 Original Notes adopted from April...

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