Topology2009Aug - T OPOLOGY QUALIFYING EXAM AUGUST 2009...

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TOPOLOGY QUALIFYING EXAM AUGUST 2009 There are six problems. Begin your answer to any problem on a new page in your blue book(s). Make a space between answers to separate parts ofa question tofacilitate grading. All answers must be justified with proofs. The relative point values are indicated. 1 (12 pts) Let Y be an ordered set in the order topo10gy. Let f, g : X ~ Y be continuous. a) Show that the set {x I f(x);:::: g(x)} is closed in X. b) Let h : X ~ Y be the function hex) = max{f(x), g(x)}. Show that h is continuous. 2 (18 pts) a) Show if Y is Hausdorff, then y X (= the space of all functions from X to Y) with the compact open topology is also Hausdorff. b) Let /" : R ~ R (R = the real numbers with the usual topology) be defined by 1 I 2 2 --yn -x if[xl < n n o iflxl;::::n Show that {fJdoes not converge to the constant function g(x) = 1 in the uniform topology, but that it does converge to this constant function in the compact convergence topology. c) Let
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Topology2009Aug - T OPOLOGY QUALIFYING EXAM AUGUST 2009...

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