Download BBA IInd SEMESTER EXAMINATION 2008-09

M.Sc. Maths III (Third) Semester Examination 2013-14
Course Code: MAT301
Paper ID: 0503215
Topology
Time: 3 Hours
Max. Marks: 70
Note: Attempt six questions in all. Q. No. 1 is compulsory.
1.
a)
b)
c)
d)
e)
f)
g)
h)
Answer any five of the following (limit your answer in 50
words).
(4x5=20)
In any topological space, prove that A  D(A) is closed.
Prove that the family of all open subsets of a metric space
defines a topology on it.
Let (x,d) be a pseudo metric space and let B be the collection
of all open spheres  ( x, r ), then show that B is a base far
some topology on x.
Show by mean of counter example that seprability is not
hereditary property.
Let P(X,T)  (Y,U) be a continuous map. Show that
i)
if π is finer than T2f is T1-U continuous.
ii)
U1 is coarser than U1 then f is T-U1 continuous.
Prove that every completely normal space is normal.
Let (X,T) be a topological space and ACX. Let G  H be a
disconnection of A, then show that A A  G.A  H are
separated sets.
A topological space (X,T) is a Housdorff space iff every
convergent filter on X has a unique limit.
Let (X,d) be a metric – space. If <xn> and <yn> are sequences
in X, such that xn  ,yn  y, then show that d(xn,yn)
(5)
 d(x,y). .
b) Let (X,d) be metric – space and A  X. Show that
A  {x  x : d ( x, A)  0}.
2. a)
3. a)
Prove that every convergent sequence in a metric space is a
Cauchy-sequence.
(5)
b) Define interior points of a topological space. Prove that
i)
( A  B)   A   B 
ii)
A   B   ( A  B) 
(5)
4. a) Define Homeomorphism, Let (x,z) and (y,v) be topological
spaces and let f be a bijective mapping of x to y. Then the
following statements are equivalent:
(5)
i)
f is a homeomorphism
ii)
f is continuous and open
b) Prove that Let (x,z and y,v) be two topological spaces. A
mapping f : x  y is closed iff f [ A ]  ( f [ A]) for every
A  X ).
(5)
5. a) Define compactness in a topological space. Prove that let f be
a continuous mapping of a compact topological space x into a
topological space y, then f[x] is compact.
(5)
b) Define T2-space give an example to show that a T1-space
head not be T2.
(5)
6. a) Define normal space. Prove that every compact Houdroff
space in normal (T4).
(5)
b) Product that the product space X  Y is compact iff each of
the spaces XD and Y is compact.
(5)
7. a) Define a metrizable topological space with an example. (5)
b) Prove that the topological product of a finite family of
metrizable space is metrizable.
(5)
8.
State and prove Tietze extension theorem.
(10)