LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

CV 10

THIRD SEMESTER – APRIL 2006

                                                                  MT 3800 – TOPOLOGY

 

 

Date & Time : 28-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

 

 

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  is also a metric on X.  Give an example of a pseudo metric which is not a metric.

(or)

1. ii) In any metric space X, show that each open sphere is an open set.  Prove that any union of open sets in X is open.                                                                                                                 (8)
2. 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.
3. ii) State and prove Cantor’s Intersection Theorem.

iii)  If  is a sequence of nowhere dense sets in a complete metric space X, prove that there exists a point in X which is not any of the s.                                                               (6+6+5)

1. iv) Let X and Y be metric spaces and f be a mapping of X into Y.  Then prove that f is continuous iff is open in X whenever G is open in Y.
2. v) Prove that the set C(X,R) of all bounded continuous real functions defined on a metric space X is a Banech space with respect to point wise addition and scalar multiplication and the norm defined by .                                                                                                                  (6+11)
3. a) i) Show that every separable metric space is second countable.

(or)

1. ii) Prove that the product of any non-empty class of compact spaces is compact.

(8)

1. b) i) Show that any continuous image of a compact space is compact.
2. 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+6+5)

(or)

1. State and prove Lindelof’s Theorem.
2. v) Let X be any non-empty set, and let S be an arbitrary class of subsets of X.  Show that S can serve as an open subbase for a topology on X.                                                            (6+11)

III. a) i)   Prove that a metric space is compact iff it is complete and totally bounded.

(or)

1. ii) Prove that every compact Hausdorff space is normal.                                     (8)
2. b) i) In a sequentially compact metric space, prove that every open cover has a Lebesque number.
3. ii) Show that every sequentially compact metric space is totally bounded.

iii)  Prove that every sequentially compact metric space is compact.             (9+4+4)

(or)

1. b) iv) In a Hausdorff space, show that any point and disjoint compact subspace can be separated by open sets.
2. v) Show that every compact subspace of a Hausdorff space is closed.
3. vi) Prove that a 1–1 mapping of a compact space on to a Hausdorff space is homeomorphism.         (7+5+5)
4. a) i) Prove that any continuous image of a connected space is connected.

(or)

1. ii) Let X be a T₁  Prove that X is normal iff each neighbourhood of a closed set F contains the closure of some neighbourhood of F.                                                                           (8)
2. b) i) State and prove the Urysohn Imbedding Therorem.

(or)

1. ii) State and prove the Weierstrass Approximation Theorem.                            (17)

 

 

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             LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034  M.Sc. DEGREE EXAMINATION – MATHEMATICS

AA 23

THIRD SEMESTER – NOV 2006

         MT 3803 – TOPOLOGY

(Also equivalent to MT 3800)

 

 

Date & Time : 25-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

            Answer all the questions.

 

01.(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 and Y be metric spaces and f be a mapping of X into Y.

Show that f ^–1(G) is open in X whenever G is open in Y.

 

(b)(i)  Let X be a metric space.  Prove that any arbitrary union of open sets in X is
open and any finite intersection of open sets in X is open.

 

(ii)   Give an example to show that any arbitrary intersection of open sets in X
need not be open.

 

(iii)   In any metric space X, prove that each closed sphere is a closed set.(6+4+5)

(OR)

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

 

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

 

(vi)   If {A_n} is a sequence of nowhere dense sets in a complete metric space X,
show that there exists a point in X which is not in any of the A_n’s.    (4+6+5)

 

02.(a) (i)    Prove that every separable metric space is second countable.

(OR)

(ii)   Let X be a non–empty set, and let there be give a “closure” operation
which  assigns to each subset A of X a subset   of X in such a manner
that  (1)  = ,  (2)  A Í ,  (3)  , and  (4)   =.

If a “closed” set A is defined to be one for which A = , show that the
class of all complements of such sets is a topology on X whole closure
operation is precisely that initially given.

 

• (i) Show that any closed subspace of a compact space is compact.

 

(ii)   Give an example to show that a proper subspace of a compact space need
not be closed.

 

(iii)  Prove that any continuous image of a compact space is compact.     (5+4+6)

(OR)

(iv)  Let C(X  đ) be the set of all bounded continuous real functions defined
on a topological space X.  Show that  (1) C (X  đ) is a real Banach space
with respect  to pointwise addition and multiplication and the  norm
defined by  = sup;    (2)  If multiplication is defined pointwise
C(X,  R) is a commutative real algebra with identity in which
£     and   = 1.

 

03.(a) (i)     State and prove Tychonoff’s Theorem.

(OR)

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

 

(b) (i)    Prove that in a sequentially compact space, every open cover has a
Lesbesgue number.

 

(ii)    Show that every sequentially compact metric space is totally bounded.(9+6)

(OR)

(iii)    State and prove Ascoli’s Theorem.

 

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

(OR)

(ii)     Prove that every compact Haurdolff space is normal.

 

(b)(i)     Let X be a T₁ – space.

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

 

(ii)    State and prove Uryjohn’s Lemma.                                                        (6+9)

(OR)

(iii)    If X is a second countable normal space, show that there exists a
homeomorphism  f  of X onto a subspace of R^¥_.

 

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

(OR)

(ii)    Show that the components of a totally disconnected space are its points.

 

(b)(i)    Let X be a topological space and A be a connected subspace of X.  If B is a
subspace of X such that A Í B Í , show that B is connected.

 

(ii)   If X is an arbitrary topological space, then prove the following:

(1)  each point in X is contained in exactly one component of X;

• each connected subspace of X is contained in a component of X;
• a connected subspace of X which is both open and closed is a            component of X.                                                                        (6+9)

(OR)

 

(iii)   State and prove the Weierstrass Approximation Theorem.

 

 

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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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    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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

MT 3810 – TOPOLOGY

 

 

Date : 21-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

Answer all questions. All questions carry equal marks.                            5 x 20 = 100 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 conditions. a) b)                   Show that d is a metric on X.             (or)
		(ii)	Let X be a metric space. Prove that a subset G of X is open it is a union of open spheres.                                                                                                                          (5)
	(b)	(i)	Let X be a metric space, and let Y be a subspace of X. Prove that Y is complete iff Y is closed.
		(ii)	State and prove Cantor’s Intersection Theorem.
		(iii)	State and prove Baire’s Theorem.                                                                             (6+5+4) (or)
		(iv)	Let X and Y be metric spaces and let f be a mapping of X into Y. Prove that f is continuous at   and f is continuous is open in X whenever G is open in Y.                                                                                           (15)
02)	(a)	(i)	Prove that every separable metric space is second countable. (or)
		(ii)	Define a topology on a non-empty set  with an example. Let  be a topological space and  be an arbitrary subset of . Show that each neighbourhood of intersects .                                                                                                              (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+6+3) (or)
		(iv)	Show that a topological space is compact, if every subbasic open cover has a finite subcover. (15)
03)	(a)	(i)	State and prove Tychnoff’s Theorem. (or)
		(ii)	Show that a metric space is compact if it is complete and totally bounded.                   (5)
	(b)	(i)	Prove that in a sequentially compact space, every open cover has a Lebesgue’s number.
		(ii)	Prove that every sequentially compact metric space is totally bounded.                    (10+5)                                     (or)
		(iii)	State and prove Ascoli’s Theorem.                                                                                (15)
04)	(a)	(i)	Show that every subspace of Hausdorff space is also Hausdorff. (or)
		(ii)	Prove that every compact Hausdorff Space is normal.                                                   (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)	If X is a second countable normal space, prove that there exists a homeomorphism f of X onto a subspace of and X is therefore metrizable.                                               (15)
05)	(a)	(i)	Prove that any continuous image of a connected space is connected. (or)
		(ii)	Show that the components of a totally disconnected space are its points.                        (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 the Weierstrass Approximation Theorem.                                             (15)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

MT 3810 / 3803 – TOPOLOGY

 

 

Date : 01/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

Answer all questions. All questions carry equal marks:                      5 x 20 = 100

01.	(a)	(i)	Let X be a metric space with metric . Show that defined by is also a metric on X. (OR)                                                                                                                              (OR)
		(ii)	Define a Pseudo metric space on a non-empty set X. Give an example of a pseudo metric which is not a metric.               (5)
	(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)	Prove that f is continuous at .
		(iv)	Show that  f is continuous is open in X whenever G is open in Y.
02.	(a)	(i)	Prove that every second countable space is separable. (OR)                                                                                                                              (OR)
		(ii)	Define a topology on a non-empty set  with an example. Let  be a topological space and  be an arbitrary subset of . Show that each neighbourhood of intersects .                                                                                                 (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+6+3) (OR) (OR)
		(iv)	Show that a topological space is compact, if every subbasic open cover has a finite subcover. (15)
03.	(a)	(i)	Show that every compact metric space has the Bolzano-Weirstrass property. (OR)                                                                                                                              (OR)
		(ii)	State and prove Tychanoff’s Theorem.         (5)
	(b)	(i)	Prove that In a sequentially compact metric space every open cover has a Lebesgue number.
		(ii)	Show that every sequentially compact metric space is compact. (10+5) (OR)                                                                                                                              (OR)
		(iii)	State and prove Ascoli’s Theorem               (15)
04	(a)	(i)	Show that the product of any non-empty class of Hausdorff spaces is a Hausdorff spaces. (OR)                                                                                                                              (OR)
		(ii)	Prove that every compact Haurdorff space is normal.                           (5)                                                                                        (5)
	(b)	(i)	Let X be a T1 – space. Show that X is a normal  each neighbourhood of a closed set F contains the closure of some neighbourhood of F.
		(ii)	State and prove Urysohn’s Lemma.                (6+9)                                                                                                                                                                                                                                                                                                                       (6+9)                                                                     (OR)                                                                                                                              (OR)
		(iii)	If  X is a second countable normal space, show that there exists a homeomorphism f of X onto a subspace of .                    (15)                                                                                                                                (15)
05.	(a)	(i)	Show that any continuous image of a connected space is connected. (OR)                                                                                                                              (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)                                                                                                                                                      (6+9) (OR)                                                                                                                              (OR)
		(iii)	State and prove Weierstrass Approximation Theorem.           (15)                        (15)

 

 

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