LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AB 32

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – November 2008

    MT 3803 – TOPOLOGY

 

 

 

Date : 03-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

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

d₁(x,y)  =   is also a metric on  X.

(OR)

(ii)  Let X be a metric space. Prove that a subset F of X is closed  Û its complement
F’ is open.                                                                                                   (5)

 

(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.

(ii) State and prove Cantor’s Intersection Theorem.

 

(iii) If  {A_n} is a sequence of nowhere dense sets in a complete metric space X,
prove that there exists a point in X which is not in any of the A_n’s.

(6 + 5 + 4)

(OR)

(iv) Prove that the set C(X,  R) of all bounded continuous real functions defined
on a  metric space X is a Banach space with respect to pointwise addition and
scalar multiplication and the norm defined by                    (15)

 

II.(a) (i)  If X is a second countable space X, prove that X is separable.

 

(OR)

(ii) Let X be a topological space, and let {f_n} be a sequence of real functions
defined on X which converges uniformly to a function  f  defined on X.  If all
the f_n’s are  continuous, show that  f  is continuous.                                     (5)

 

• (i) Show that the continuous image of a compact space is compact.

 

(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 + 5 + 4)

 

(OR)

 

(iv) Prove that a topological space is compact, if every subbasic open cover has a
finite sub cover.                                                                                          (15)

 

III. (a) (i) Show that every compact metric space has the Bolzano-Weierstrass property.

 

(OR)

 

(ii) Prove that a metric space is compact  Û it is complete and totally bounded.

(5)

(b)(i) In a sequentially compact metric space, prove that every open cover has a
Lebesgue number.

 

(ii)   Show that every sequentially compact metric space is compact.

 

(OR)

 

(iii)  State and prove Ascoli’s Theorem                                                                (15)

 

IV.(a) (i)  Prove that  a 1 – 1 mapping of a compact space onto a Haurdorff space is a
homeomorphism.

 

(OR)

 

(ii) Show that every compact space is normal.                                                  (5)

 

(b)(i) State and prove the Tietze Extension Theorem.

(15)

(OR)

 

(ii) Let X be a T₁ – space.

Show that X is normal  Û each neighbourhood of a closed set F contains the
closure of some neighbourhood of F.

 

(iii) State and prove URYSOHN’s Lemma

(6 + 9)

1. (a)(i) Prove that any continuous image of a connected space is connected.

(OR)

(ii)  Let X  be a topological space and A  be a connected subspace of X.                                                                                                                                                        (5)

If B is a subspace of  X such that A Í B  Í  ,  then show that B is connected.

 

(b)(i)  Show that a topological space X is disconnected  Û  there exists a continuous
mapping of X onto the discrete two-point space  {0, 1}.

 

(ii)  Prove that the product of any non-empty class of connected spaces is
connected.                                                                                               (6 + 9)

 

(OR)

 

(iii) State and prove the Weierstrass Approximation Theorem.                          (15)

 

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