Download M 925 - Loyola College

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Geometrization conjecture wikipedia, lookup

Grothendieck topology wikipedia, lookup

Fundamental group wikipedia, lookup

Covering space wikipedia, lookup

Continuous function wikipedia, lookup

3-manifold wikipedia, lookup

Brouwer fixed-point theorem wikipedia, lookup

General topology wikipedia, lookup

Transcript
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2008
KZ 71
M 925 - TOPOLOGY
Date : 05-05-08
Time : 9:00 - 12:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions.:
1. a)
(4 x 25 = 100 marks)
(i) Show that f : X  Y is continuous iff for each x  X and each 
neighbourhood V of f (x), there exists a neighbourhood  of x such that f ()  V .
(ii) Let X and Y be topological spaces and let f : X  Y . Show that the following statements are
equivalent:
1) f is continuous.
2) For every subset A of X, one has f (A)  f ( A)
3) For every closed set B in Y, the set f 1 ( B) is closed in X. (8)
b)
2. a)
b)
3. a)
b)
4. a)
b)
(i) Show that the space R  in the products topology is metrizable. (17)
(OR)
(ii) Let f : X  Y be a function where X is metrizable. Show that the
function f is continuous iff for each convergent sequence xn  x in X , the sequence f ( xn )
converges to f ( x )
(iii) State and prove Uniform Limit theorem. (8+9)
(i) Prove that every compact subset of a Hausdorff space is closed.
(OR)
(ii) Let X be a nonempty compact Hansdorff space. If every point of X is a
limit point of X, prove that X is uncountable. (8)
(i) If L is a linear continuous in the order topology, prove that L is connected and so is every
interval and ray in L.
(OR)
(ii) Let X be a metrizoble space. Show that the following are equivalent:
1) X is compact.
2) X is limit point compact
3) X is sequentially compact. (17)
(i) Show that a product of regular space is regular
(OR)
(ii) Show that a regular space with a countable basis is normal. (8)
(i) State and prove Tietze extension theorem.
(OR)
(ii) State and prove Uryshon’s metrizetion theorem. (17)
(i) Show that the space Y  is complete in the uniform metric where the
space Y is complete.
(OR)
(ii) State and prove extension property of Stone Cech compactification (8)
(i) State and prove Alcoli’s theorem
(OR)
(ii) State and prove Tychonoff’s theorem. (17)