LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
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THIRD SEMESTER – APRIL 2006
MT 3800 – TOPOLOGY
Date & Time : 28-04-2006/1.00-4.00 P.M. Dept. No. Max. : 100 Marks
Answer ALL questions. All questions carry equal marks.
- a) i) Let X be a metric space with metric d. Show that d1 defined by is also a metric on X. Give an example of a pseudo metric which is not a metric.
(or)
- ii) In any metric space X, show that each open sphere is an open set. Prove that any union of open sets in X is open. (8)
- b) i) Let X be a complete metric space and let Y be a subspace of X. Prove that Y is complete iff it is closed.
- ii) State and prove Cantor’s Intersection Theorem.
iii) If is a sequence of nowhere dense sets in a complete metric space X, prove that there exists a point in X which is not any of the s. (6+6+5)
- iv) Let X and Y be metric spaces and f be a mapping of X into Y. Then prove that f is continuous iff is open in X whenever G is open in Y.
- v) Prove that the set C(X,R) of all bounded continuous real functions defined on a metric space X is a Banech space with respect to point wise addition and scalar multiplication and the norm defined by . (6+11)
- a) i) Show that every separable metric space is second countable.
(or)
- ii) Prove that the product of any non-empty class of compact spaces is compact.
(8)
- b) i) Show that any continuous image of a compact space is compact.
- ii) Prove that any closed subspace of a compact space is compact.
iii) Give an example to show that a compact subspace of a compact space need not be closed. (6+6+5)
(or)
- State and prove Lindelof’s Theorem.
- v) Let X be any non-empty set, and let S be an arbitrary class of subsets of X. Show that S can serve as an open subbase for a topology on X. (6+11)
III. a) i) Prove that a metric space is compact iff it is complete and totally bounded.
(or)
- ii) Prove that every compact Hausdorff space is normal. (8)
- b) i) In a sequentially compact metric space, prove that every open cover has a Lebesque number.
- ii) Show that every sequentially compact metric space is totally bounded.
iii) Prove that every sequentially compact metric space is compact. (9+4+4)
(or)
- b) iv) In a Hausdorff space, show that any point and disjoint compact subspace can be separated by open sets.
- v) Show that every compact subspace of a Hausdorff space is closed.
- vi) Prove that a 1–1 mapping of a compact space on to a Hausdorff space is homeomorphism. (7+5+5)
- a) i) Prove that any continuous image of a connected space is connected.
(or)
- ii) Let X be a T1 Prove that X is normal iff each neighbourhood of a closed set F contains the closure of some neighbourhood of F. (8)
- b) i) State and prove the Urysohn Imbedding Therorem.
(or)
- ii) State and prove the Weierstrass Approximation Theorem. (17)