LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

ZA 60

THIRD SEMESTER – April 2009

MT 3803 / 3800 – TOPOLOGY

 

 

 

Date & Time: 16/04/2009 / 1:00 – 4:00         Dept. No.                                                       Max. : 100 Marks

 

 

            Answer ALL questions.  All questions carry equal marks.

 

01.(a) (i)   Let X be a non-empty set, and let  d  be  a  real function of ordered pairs of
elements of X which satisfies the following two conditions:

 

• d(x, y) = 0 Û x = y
• d(x, y) £ d(x, z) + d(y, z)

Show that  d  is a metric on X.

 

(OR)

(ii)  In any metric space, show that

• any union of open sets in X is open
• any finite intersection of open sets in X is open. (5)

 

(b) (i)   If a convergent sequence in a metric space has infinitely many distinct points,
then prove that its limit is a limit point of the set of points of the sequence.

 

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

 

(iii) State and prove Baire’s Theorem.                                                   (5 + 5 + 5)

 

(OR)

(iv) Proving the necessary lemmas, establish that the set  Rⁿ of all n-tuples
x = (x,₁, x₂, …,x_n) of real numbers is a real Banach space with respect to
coordinatewise addition and scalar multiplication and the norm
defined by                                                                    (15)

 

II.(a)  (i)  Show that every separable metric space is second countable.

 

(OR)

(ii)  If  f  and  g  are continuous real functions defined on a topological space X,
prove that  fg is continuous.                                                                                    (5)

 

(b)  (i) Show that any 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 subcover.                                                                                         (15)

 

III.(a) (i)  Prove that a metric space is sequentially compact  Û it has the
Bozano-Weierstrass property.

 

(OR)

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

(5)

(b)(i)  State and prove Lebesgue’s covering Lemma.

 

(ii)  Prove that every sequentially compact metric space is compact         (10 + 5)

(OR)

 

(iii)  If X is a compact metric space, then prove that a closed subspace of C(X,  R) is
compact  Û  it is bounded and equicontinuous.

(15)

IV.(a) (i) Prove that the product of any non-empty class of Hausdorff spaces is a
Hausdorff space.

(OR)

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

 

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

 

(OR)

(ii) State and prove the Urysohn Imbedding Theorem                                      (15)

 

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

 

(OR)

(ii)  Let X be a  topological space.  If  {A_i} is a non-empty class of connected
subspaces of X such that  Ç A_i  is non-empty,  prove that A =  È A_i  is also a
connected  subspace of X.                                                                           (5)

 

(b)(i) Prove that a subspace of the real line    R  is connected  Û it is an interval.

 

(ii) Let X be an arbitrary topological space.   Show that each point in X is contained
in exactly one component of X.                                                               (9 + 6)

 

(OR)

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

 

 

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