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)

 

 

Go To Main page

 

             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.

 

 

Go To Main page