LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 10

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.

1. 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)

1. 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)
2. 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.
3. 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)

1. 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.
2. 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)
3. a) i) Show that every separable metric space is second countable.

(or)

1. ii) Prove that the product of any non-empty class of compact spaces is compact.

(8)

1. b) i) Show that any continuous image of a compact space is compact.
2. 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)

1. State and prove Lindelof’s Theorem.
2. 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)

1. ii) Prove that every compact Hausdorff space is normal.                                     (8)
2. b) i) In a sequentially compact metric space, prove that every open cover has a Lebesque number.
3. 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)

1. b) iv) In a Hausdorff space, show that any point and disjoint compact subspace can be separated by open sets.
2. v) Show that every compact subspace of a Hausdorff space is closed.
3. vi) Prove that a 1–1 mapping of a compact space on to a Hausdorff space is homeomorphism.         (7+5+5)
4. a) i) Prove that any continuous image of a connected space is connected.

(or)

1. 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)
2. b) i) State and prove the Urysohn Imbedding Therorem.

(or)

1. ii) State and prove the Weierstrass Approximation Theorem.                            (17)

 

 

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