Download BBA IInd SEMESTER EXAMINATION 2008-09

M.Sc. Maths III (Third) Semester Examination 2015-16
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)
Prove that if A is open in a topological space than it is
neighbourhood of all of its points.
If X  {a, b, c, d , e} and   { , X ,{a},{a, b},{a, c, d },{a, b, e},
{a, b, c, d }} is a topology on x then find the boundary points
of A  {a, b, c}.
Prove or disprove that a constant function f : X  Y where
X and Y are topological spaces, is continuous.
Prove or disprove that subspace of a connected space is
connected.
Show that subspace of a Hausdorff space is Hausdorff.
If a is a compact subset of a Hausdorff space x then show that
if a  A then there is a open set G such that a  G  AC .
Show that every component is closed.
Define homotopy. Show that the relation of homotopy is
symmetric.
2.
State and prove Cantor’s intersection theorem.
(10)
3.
Fid all the topologies on X={a, b, c}.
(10)
4.
If X and Y are topological spaces and f : X  Y then prove
that the following are equivalent:
(10)
a) f is continuous
b) For every subset A of X, f ( A )  f ( A)
c) For every closed subset B of Y, f 1 ( B) is closed in X
d) For each x  X and each neighbourhood v of f(x) there is a
neighbourhood u of x such that f (u )  v.
5.
If A is a connected subspace of X and if A  B  A then B
is also connected.
(10)
6.
If X is a fist countable space then prove that X is Hausdorff if
and only if every convergent sequence has a unique
limit.
(10)
7.
State and prove Urysohn’s metrization theorem.
8.
Prove that the inclusion map J : S n  R n1  {0} induces an
isomorphism of fundamental groups.
(10)
(10)