Download M 925 - Loyola College

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)