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# Topology2008aug - August 2008 Qualifying Examination...

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August 2008 Qualifying Examination Topology Problems 1-4 consist of true or false statements. Each statement is to be proved or disproved with brief but complete reasoning. Provide definitions of all underlined, italicized words and phrases. On page 2 find definitions and notations of some items appearing in the problems. There are 5 problems in total. Spacing: begin each problem on a new page. 1 a) Any infinite set with the finite complement topology is connected. b) The subspace of , where is connected. c) The real line with the lower limit topology is connected. d) The complement of the “equatorial ” in the Mobius band is connected. 2 a) The set of integers with the topology generated by the basis is compact. b) The space defined in 2 a) is limit point compact . c) The set of all real invertible matrices is compact. d) The set of all real orthogonal matrices is compact. 3 a) Define a function from onto a three point set by With the resulting quotient topolog y induced by is Hausdorff. b) For any map

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Topology2008aug - August 2008 Qualifying Examination...

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