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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 M.Sc DEGREE EXAMINATION – MATHEMATICS THIRD SEMESTER – APRIL 2008 KZ 71 M 925 - TOPOLOGY Date : 05-05-08 Time : 9:00 - 12:00 Dept. No. Max. : 100 Marks PART – A Answer ALL questions.: 1. a) (4 x 25 = 100 marks) (i) Show that f : X Y is continuous iff for each x X and each neighbourhood V of f (x), there exists a neighbourhood of x such that f () V . (ii) Let X and Y be topological spaces and let f : X Y . Show that the following statements are equivalent: 1) f is continuous. 2) For every subset A of X, one has f (A) f ( A) 3) For every closed set B in Y, the set f 1 ( B) is closed in X. (8) b) 2. a) b) 3. a) b) 4. a) b) (i) Show that the space R in the products topology is metrizable. (17) (OR) (ii) Let f : X Y be a function where X is metrizable. Show that the function f is continuous iff for each convergent sequence xn x in X , the sequence f ( xn ) converges to f ( x ) (iii) State and prove Uniform Limit theorem. (8+9) (i) Prove that every compact subset of a Hausdorff space is closed. (OR) (ii) Let X be a nonempty compact Hansdorff space. If every point of X is a limit point of X, prove that X is uncountable. (8) (i) If L is a linear continuous in the order topology, prove that L is connected and so is every interval and ray in L. (OR) (ii) Let X be a metrizoble space. Show that the following are equivalent: 1) X is compact. 2) X is limit point compact 3) X is sequentially compact. (17) (i) Show that a product of regular space is regular (OR) (ii) Show that a regular space with a countable basis is normal. (8) (i) State and prove Tietze extension theorem. (OR) (ii) State and prove Uryshon’s metrizetion theorem. (17) (i) Show that the space Y is complete in the uniform metric where the space Y is complete. (OR) (ii) State and prove extension property of Stone Cech compactification (8) (i) State and prove Alcoli’s theorem (OR) (ii) State and prove Tychonoff’s theorem. (17)