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M.Sc. Maths III (Third) Semester Examination 2015-16 Course Code: MAT301 Paper ID: 0503215 Topology Time: 3 Hours Max. Marks: 70 Note: Attempt six questions in all. Q. No. 1 is compulsory. 1. a) b) c) d) e) f) g) h) Answer any five of the following (limit your answer in 50 words). (4x5=20) Prove that if A is open in a topological space than it is neighbourhood of all of its points. If X {a, b, c, d , e} and { , X ,{a},{a, b},{a, c, d },{a, b, e}, {a, b, c, d }} is a topology on x then find the boundary points of A {a, b, c}. Prove or disprove that a constant function f : X Y where X and Y are topological spaces, is continuous. Prove or disprove that subspace of a connected space is connected. Show that subspace of a Hausdorff space is Hausdorff. If a is a compact subset of a Hausdorff space x then show that if a A then there is a open set G such that a G AC . Show that every component is closed. Define homotopy. Show that the relation of homotopy is symmetric. 2. State and prove Cantor’s intersection theorem. (10) 3. Fid all the topologies on X={a, b, c}. (10) 4. If X and Y are topological spaces and f : X Y then prove that the following are equivalent: (10) a) f is continuous b) For every subset A of X, f ( A ) f ( A) c) For every closed subset B of Y, f 1 ( B) is closed in X d) For each x X and each neighbourhood v of f(x) there is a neighbourhood u of x such that f (u ) v. 5. If A is a connected subspace of X and if A B A then B is also connected. (10) 6. If X is a fist countable space then prove that X is Hausdorff if and only if every convergent sequence has a unique limit. (10) 7. State and prove Urysohn’s metrization theorem. 8. Prove that the inclusion map J : S n R n1 {0} induces an isomorphism of fundamental groups. (10) (10)