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325 HW 06

# 325 HW 06 - MAT 325 Topology Professor Zoltan Szabo Problem...

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MAT 325: Topology Professor Zoltan Szabo Problem Set 6 Rik Sengupta [email protected] March 30, 2010 1. Munkres, p. 199, problem 3 Show that every order topology is regular. Solution. We first prove that every order topology is Hausdorff. So let ( X, ) be a simply ordered set. Let X be equipped with the order topology induced by the simple order. Fur- thermore, let a and b be two distinct points in X , and suppose without loss of generality that a < b . Let A = { x X : a < x < b } , i.e. the set of elements between a and b . So now, if A is empty, then a ( -∞ , b ), b ( a, ), and ( -∞ , b ) ( a, ) = , and so X is Hausdorff. If A is nonempty, then a ( -∞ , x ), b ( x, ), and ( -∞ , x ) ( x, ) = for any x A , and therefore, X is Hausdorff. So in particular, single points in X are closed. Suppose now that x X , and A is a closed set, disjoint from x . Then, there exists a basis element ( a, b ) containing x which is disjoint from A . So pick any a 0 ( a, x ), and let U 1 = ( -∞ , a 0 ), V 1 = ( a 0 , ). If no such a 0 exists, then let U 1 = ( -∞ , x ), V 1 = ( a, ). Exactly as before, in both cases, the pair of sets is disjoint. Similarly, try to find b 0 ( x, b ), and if that exists, let U 2 = ( b 0 , ), V 2 = ( -∞ , b 0 ), and if not, let U 2 = ( x, ), V 2 = ( -∞ , b ).

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325 HW 06 - MAT 325 Topology Professor Zoltan Szabo Problem...

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