Tamil Nadu Open University (TNOU) 2009-1st Year M.Sc Mathematics Tamilnadu open university Maths Topology and measure theory - Question Paper

M.Sc. DEGREE EXAMINATION
JUNE 2009.

First Year

(AY 2005-06 and CY 2006 batches only)

Mathematics

TOPOLOGY AND MEASURE THEORY

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

Each question carries 5 marks.

1.         Let X and Y be topological spaces; let . Then the following are equivalent :

             (a) f is continuous

             (b) for every subset A of X, one has .

             (c) for every closed set B in Y, the set is closed in X.

2.         Prove that the union of a collection of connected sets that have a point in common is connected.

3.         Prove that compact subset of a Hausdorff space is closed.

4.         Suppose that X has a countable basis. Then prove that (a) every open covering of X contains a countable subcollection covering X. (b) There exists a countable subset of X which is dense in X.

5.         Let be a countable collection of sets of real numbers. Then prove that .

6.         State and prove the bounded convergence theorem.

7.         If and then prove that .

8.         If f and g are measurable functions then prove that and are measurable.

PART B (5 10 = 50 marks)

Answer any FIVE questions.

Each question carries 10 marks.

9.         Let be the standard bounded metric on R. If and y are two points of R^w, define . Then prove that D is a metric that induces the product topology on R^w.

10.       Prove that the Cartesian product of connected spaces is connected.

11.       (a) State maximum and minimum value theorem. (4)

             (b) Let X be a compact Hausdorff space. If every point of X is a limit point of X then prove that X is uncountable. (6)

12.       (a) Prove that every compact Hausdorff space is normal. (4)

             (b) Prove that every regular space with a countable basis is normal. (6)

13.       Prove that the collection M of all measurable sets form a algebra.

14.       Let f be defined and bounded on a measurable
set E with mE finite. Prove that

             for all simple functions and , it is necessary and sufficient that f be measurable.

15.       (a) State and prove the Fatou's lemma for the general measure. (6)

             (b) State and prove the Lebesgue convergence theorem. (4)

16.       State and prove the Radon-Nikodym theorem.