MA/MSCMT-02
June Examination 2016
M.A. M.Sc. (Previous) Mathematics Examination
Real Analysis and Topology
Paper MA/MSCMT-02
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C. Write answer as per the given instructions.
Section A 8 × 2 16
(Very Short Answer Questions)
Note: Section contain 08 Very Short Answer Type Questions. Examinees have to attempt all questions. Each Question is of 02 marks and maximum word limit may be thirty words.
Define σ Algebra.
Define outer measure of a set.
Define Borel measurable function.
State Fatou' lemma for measurable function.
Define Hilbert space.
231
MA/MSCMT-02 3000 4 (P.T.O.)
MA/MSCMT-02 3000 4 (Contd.)
231
Define neighbourhood of a point x.
Define local base at any point of a Topological space.
(viii)Write finite intersection property for compact spaces.
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contain 08 Short Answer Type Questions.
Examinees will have to answer any Four questions.
Each question is of 08 marks. Examinees have to delimit
each answer in maximum 200 words.
Prove that a non-empty collection R of subsets of a set X is an
algebra of sets If and only If R is a ring of sets and X R.
Prove that the collection M of measurable sets is a σ algebra.
Prove that If fn is a convergent sequence of measurable
functions defined on measurable set then the limit function
of fn is measurable.
State and prove Minkowski's inequality.
Let D 1 L2 be everywhere dense in L2. If Parseval's identity
holds for all functions in then prove that the system is
closed.
Prove that a second countable space is always first countable
but converse is not true.
MA/MSCMT-02 3000 4 (P.T.O.)
231
Prove that a topological space is a normal space iff for any
closed set F and an open set G containing there exist an open
set V such that F 1 V 1 V 1 G.
Prove that continuous image of a connected space is connected.
Section C 2 × 16 32
(Long Answer Type Questions)
Note: Section contain 04 Long Answer Type Questions.
Examinees will have to answer any Two questions.
Each question is of 16 marks. Examinees have to delimit
each answer in maximum 500 words. Use of nonprogrammable
scientific calculator is allowed in this paper.
10) Prove that a topological space X is hausdroff space Iff
every convergent filter on X has unique limit.
State and prove Alexander subbase lemma for compact
topological space.
11) Let R be the set of all real numbers show that the
set S yh x2 y2 in R2 is the one point
compactification of R and that ∞ is the point at
infinity.
Prove that every closed subspace of a locally compact
space is locally compact.
MA/MSCMT-02 3000 4
231
12) If and be two non-negative measurable functions
of the set If then prove that
h(x)dx f(x)dx g(x)dx
E E E
w w w
State and prove Cauchy-Bunyakowski-Schwarty in
equality for space L2.
13) Prove that the necessary and sufficient condition for a bounded
function f defined on interval to be L-integrable over
is that for given there exists a measurable partition
P of such that