Padmashree Dr DY Patil Vidyapeeth 2007 M.Sc Mathematics Mathamtics - exam paper

M.Sc. DEGREE EXAMINATION, 2007

( MATHEMATICS )

( 1st YEAR )

( PAPER - V )

550. SET TOPOLOGY

( New Regulations )

May ] [ Time : three Hours

Maximum : 100 Marks

PART ? A (8 × five = 40)

ans any 8 ques..

every ques. carries 5 marks.

1. Let X be a metric space. Prove that a subset G of

X is open if and only if, it is a union of

open spheres.

2. Let X be a metric space. Prove that any

intersection of closed sets in X is closed.



3. Let A be an arbitrary subset of topological space

X. Prove that

(i)

and (ii)

4. State and prove Lindelof?s theorem.

5. Prove that any closed subspace of a compact

space is compact.

6. Prove that every sequentially compact metric

space is compact.

7. Prove that the product of any non-empty class of

Hausdorff spaces is a Hausdorff space.

8. Prove that every compact Hausdorff space is

normal.

9. Prove that any continuous image of a connected

space is connected.

10. Prove that the components of a totally

disconnected space are its points.



PART ? B (3 × 20 = 60)

ans any 3 ques..

every ques. carries TWENTY marks.

11. (a) Let X and Y be metric spaces and f a

mapping of X into Y. Prove that f is

continuous if and only if, F? 1(G) is open in

X wherever G is open in Y.

(b) State and prove Cauchy?s inequality.

12. (a) State and prove Kuratowski closure axioms.

(b) Prove that every separable metric space is

second countable.

13. (a) Prove that every closed and bounded

subspace of the real line is compact.

(b) Prove that every compact metric space has

the Bolzano ? Weierstrass property.

14. State and prove Urysohn imbedding theorem.

15. State and prove Weierstrass approximation

theorem.

Earning:   Approval pending.