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PGMT-6A (PA/10/VIA)
PGMT-6A (PA/10/VIA)
POST-GRADUATE COURSE
f)
Show that union of finite number of
compact sets is compact in a topological
space. Is the union of an infinite number of
compact sets a compact set ? Give reasons.
g)
Construct a uniformity v for the space R of
reals which induces the usual topology
of R.
a)
State the Kuratowski closure axioms. Prove
that the Kuratowski closure operator
C : ℘ (X )  ℘ (X ) generates a topology  on
Assignment — June, 2017
MATHEMATICS
Paper - 6A : General Topology
Full Marks : 50
Weightage of Marks : 20%
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 weightage for each question has been
indicated in the margin.
2.
Answer Question No. 1 and any four from the rest.
1.
Answer any five questions :
a)
X such that C (A )   -closure of A for all
2  5 = 10
A  X where X is the underlying set.
In a topological space if for any two open
sets U and V we have U  V   then show
b)
suitable topological spaces so that f is
closed and continuous but not open.
d)
Show that an infinite discrete space is
locally compact without being compact.
e)
Prove that the set of all irrotational
numbers with the usual topology induced
from the usual topology of reals is totally
disconnected.
PG-Sc.-AP-3108
[ P.T.O.
3.
2
c)
Prove that the real number space R with
the lower limit topology is separable but is
not second countable.
3
a)
Prove that a mapping f : (X , )  (Y ,  ) is
closed and open.
Give
an
example
of
a
mapping

f : (X , )  (Y ,  ) where (X ,  ), (Y , ) are
In a topological space (X ,  ) show that if
G  H  G , far denotes the closure.
In a topological space (X ,  ) if A  X , show
that B dr (A )   if and only if A is both
c)
5
G is an open set in X and H is dense then
that U  A  U  V   , bar denotes closure.
b)
2
continuous f and only if f (A )  f (A ) for
every subset A of X.
b)
5
Define a net. Prove that in a topological
space (X ,  ) , a point u  X is a limit point
of A  X if and only if there is a net in
A \ { u } such that the net converges to u. 5
PG-Sc.-AP-3108
3
4.
PGMT-6A (PA/10/VIA)
a)
Prove that a topological space (X ,  ) is T1 if
and only if { x } is closed for every x  X . 3
b)
State and prove Tietze Extension theorem. 7
a)
Prove that a topological space (X ,  ) is
PGMT-6A (PA/10/VIA)
b)
4
Define a uniformity on a non-void set X.
Let (X ,v ) be a uniform space. Define
  { G  X : for each x  G , there is a
5.
compact if and only if for every family of
{F }   
closed subsets
with finite
intersection properly,

F   .
5
member U  v such that U ( x )  G } . Prove
that  is a topology on X.
1.
2.

b)
Define the one point compactification
(X u , u ) of a non-compact locally compact
T2 space (X ,  ) . Then show that (X u , u ) is
a compact T2 space.
5
6.
a)
Prove that union of a family of connected
sets no two of which are separated is also
connected.
4
b)
Prove that the continuous image of a
connected space is connected.
2
c)
Prove
that
the
product
space
is
(X Y ,    ) connected if and only if (X ,  )
and  (Y ,  ) are connected.
7.
a)
3.
4.
5.
Date of Publication
Last date of submission of
answer script by the student to
the study centre
Last date of submission of
marks by the examiner to the
study centre
Last date of submission of
marks by the study centre to
the Department of C.O.E. on or
before
Date of evaluated answer script
distribution by the study centre
to the student
4
Show that the image of a locally connected
space X under a continuous and open
mapping is also locally connected.
5
PG-Sc.-AP-3108
[ P.T.O.
PG-Sc.-AP-3108
5
: 13/02/2017
: 19/03/2017
: 16/04/2017
: 21/04/2017
: 30/04/2017