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PGMT-6A (PT/10/VIA)
PGMT-6A (PT/10/VIA)
POST-GRADUATE COURSE
e)
Term End Examination — December, 2014 / June, 2015
Paper - 6A : General Topology
Full Marks : 50
( Weightage of Marks : 80% )
Special credit will be given for accuracy and relevance
in the answer. Marks will be deducted for incorrect
spelling, untidy work and illegible handwriting.
The marks for each question has been
indicated in the margin.
Answer Question No. 1 and any four from the rest.
1.
Answer any five questions :
a)
b)
2.
f)
Prove that the set of all irrational numbers
with the usual topology induced from the
usual
topology
of
reals
is
totally
disconnected.
g)
Construct a uniformly v for the space R of
reals which induces the usual topology
of R.
a)
State the Kuratowski closure axioms.
Prove that the involved operator c ( say ):
(X )  (X ) generates a topology  ( say )
2  5 = 10
1 1
1
If A  
 1, , ,...,  , obtain limit points
n 
 2 3
of A, if any, of A if A is endowed with
(i) usual topology of reals, (ii) discrete
topology, (iii) cofinite topology.
on X such that c ( A )  closure of A in ( X ,  ) ,
for
b)
d)
Show that a finite topological space that is
T1 has discrete topology.
PG-Sc.-1310-G
[ P.T.O.
where
X
denotes
the
4
In a topological space ( X ,  ) if G is an open
and only if G  A   .
( both closed and open).
suitable topological spaces so that f is both
open and closed but not continuous.
AX,
set and A  X then show that G  A   if
In a topological space ( X ,  ) if A  X , show
Give
an
example
of
a
mapping
f : ( X ,  )  (Y ,  ) where ( X ,  ) , (Y ,  ) are
all
underlying set.
that Bdr (A )   if and only if A is clopen
c)
Prove that in a topological space ( X ,  ) the
set A consisting of the elements of a
convergent sequence along with its limit
point is compact.
MATHEMATICS
Time : 2 Hours
2
3.
3
c)
Prove that the real number space R with
the lower limit topology is separable but is
not second countable.
3
a)
Let f : ( X ,  )  (Y ,  ) . Prove that following
statements are equivalent :
i)
f is continuous.
ii)
For any closed set F in Y, f
closed in X.
PG-Sc.-1310-G
1
(F ) is
3
PGMT-6A (PT/10/VIA)
f (A )  f (A ) for every subset A of X.
iv)
f 1 (B )  f 1 (B ) for every subject B
of Y.
b)
b)
iii)
of A  X if and only if there is a net in
A \ { u } such that the net converges to u. 4
4.
5.
a)
Prove that a topological space ( X ,  ) is T2 if
and only if every net in X converges to at
most one point in X.
3
b)
State and prove Tietze Extension theorem. 7
a)
Define compactness. Is compactness a
hereditary property ? Answer with reasons.
Prove that the product space (X Y ,   )
is connected if and only if ( X ,  ) and (Y ,  )
are connected.
6
Define a net. Prove that in a topological
space ( X ,  ) , a point u  X is a limit point
7.
Prove that continuous image of a compact
space is compact.
3
c)
Define the one-point compactification
( X u , u ) of a non-compact locally compact
Prove that every component of a topological
space is closed.
1
a)
Using connectedness, prove that any
continuous function f : [ 0,1]  [ 0,1] has a
fixed point u i.e. f ( u )  u .
b)
a)
In
a
topological
space
( X, )
if
A
  {G  X
that  is a topology on X.
is
[ P.T.O.
: for each x  G , there is a
member U  v such that U ( x )  G } . Prove
connected and A  B  A then prove that
B is connected.
4
PG-Sc.-1310-G
5
Define a uniformity on a non-void set X. Let
a
uniform
space.
Define
( X ,v ) be
T2 space ( X ,  ) . Then show that ( X u , u ) is
a compact T2 space.
5
6.
5
c)
2
b)
4
PGMT-6A (PT/10/VIA)
PG-Sc.-1310-G
5