LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

XZ 42

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2008

    MT 3803 / 3800 – TOPOLOGY

 

 

 

Date : 26/04/2008            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Answer ALL questions.:                                                                               (5 X 20 = 100)

 

1. (a) (i) Let X be a metric space with metric d.

Show that d₁ defined by

is also a metric on X.

(OR)

(ii)  In any metric space X, prove that each open sphere is an open set.

(b) (i) Let X be a complete metric space and let Y be a subspace of X. Prove that

Y is complete iff Y is closed.

(ii) State and prove Cantor Intersection Theorem.                        (8+7)

(OR)

(iii) Let X and Y be metric spaces and let f be a mapping of X into Y.

Prove that f is continuous at x₀ iff  and f is

continuous iff f^-1(G) is open in X whenever G is open in Y.   (15)

 

2. (a) (i) Prove that every second countable space is reparable.

(OR)

(ii) Prove that every separable metric space is second                  (5)

(b) (i)  Show that every continuous image of a compact space is compact.

(ii) Prove that any closed subspace of a compact space is compact. (8+7)

(OR)

(c) (i)  If f and g are continuous real or complex functions defined on a metrical

space X, then show that f+g and are also continuous.

(ii) Let C(X,R) be the set of all bounded continuous real functions defined on

metric space, show that C(X,R) is a closed subset of the metric space. (8+7)

 

3. (a) (i) State and prove Tychnoff’s Theorem.

(OR)

(ii) Show that every sequentially compact metric space is compact.   (5)

(b) (i) Show that a metric space is sequentially compact it has the Bolzano

Weierstrass property.

(ii) Prove that every Compact metric space has the Bolzano Weierstrass

Property.                                                                                              (10+5)

(OR)

(iii) State and prove Lebesgue Coverity Lemme.

(iv) Prove that a metric space is compact  it is complete and totally

bounded.                                                                                              (9+6)

 

 

 

4. (a) (i) Show that every subspace of a Hausdorff space is also Hausdorff.

(OR)

(ii) Prove that every compact Hausdorff Space is normed.                         (5)

(b) (i) Prove that the product of any non-empty class of Hausdorff Spaces is a

Hausdorff Space.

(ii) Prove that every compact subspace of a Hausdorff space is closed.

(iii) Show that a one-to-one continuous mapping of a compact space onto a

Hausdorff Space is a homeomorphism.                                        (6+4+5)

(OR)

(iv) State and prove Uryshon Imbedding Theorem.                                  (15)

 

5. (a) (i) Show that any continuous image of a connected space is connected.

(OR)

Prove that if a subspace of a real line is connected, then it is an internal. (5)

(b) (i) Show that the product of any non-empty class of connected spaces is

connected.

(ii) Let X be a Compact Hausdorff Space. Show that X is totally disconnected,

iff it has open base whose sets are also closed.                                     (6+9)

(OR)

(iii) State and prove Weierstrass Approximation Theorem.

 

 

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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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