Gauhati University 2007 Previous : thetics--II - Question Paper

a A B C0 0 0 ( ) Ì Ì
b B A C0 0 0 ( ) Ì Ì
c C B A0 0 0 ( ) Ì Ì
d A B C0 0 0 ( ) = =
2007
MATHEMATICS
SECOND PAPER
(Topology and Functional Analysis)
Full Marks: 80
Time: three hours
The figures in the margin indicate full marks for the ques.
PART-A (Objective-type Questions)
(Marks: 32)
every ques. (1-16) carries 4 codes (a), (b), (c) and (d), out of which 1 is for
accurate ans. select the accurate code: 2×16=32
1. Let X be a set and Tic be the collection of all subsets U of X such that X-U is
either countable or is all of X. Then the topology Tic on X is called
a) Cofinite
b) Cocountable
c) Discrete
d) Indiscrete
2. (R, u) is the usual topological space on R. Then the closure of the set of natural
numbers (N) is
(a) R
(b) N
(c) N U{0}
(d) None
3. Let A = (a, b), B = [a, b) and C = (a, b]. Then
4. Let (R, u) be usual topological space and T be relative topology on [0, 1].
Then
(a) (½, 1] is u-open but not T-open
(b) (½, 1] is both u-open and T-open
(c) (½, 1] is neither u-open and nor T-open
(d) (½, 1] is not u-open but T-open
5. Let X = {1, 2, 3, 4, 5} and T = {
, {1}, {2, 3} {1, 2, 3}, X} is a topology on
X. Let B1 = {{1, 2}, X}, B2 = {{2, 3}, X}, B3 = {{1, 2, 3}}. Then
(a) B1 is a local base at 1
(b) B2 is a local base at 3
(c) B2 is a local base at 2
(d) B2 is not a local base at 2
T T I x a if one two ( ) Ì
a A B^ ^ ( ) =
b A B^ ^ ( ) Í
c B A^ ^ ( ) Í
( 4,0,3), (0,1,0)
5
1
(3,0,4),
5
1
(a) -
6. Let Ix: (X, T1) (X, T2) be identity map from topological space (X, T1) into
(X, T2). Then
is an open map
is a continuous map
is open
(e) None of (a), (b) and (c)
7. If R is equipped with usual topology, then
(a) (0,1) is a compact subset of R
(b) (0,1) U[1,2) is a connected set
(c) {0, 1} is a compact subset of R
(d) [0, 1) is a compact and connected set
8. (a) A Hausdorff space is metrizable
(b) A metric space is a Hausdorff space
(c) Hausdorff property is not hereditary
(d) Hausdorff is not a topological property
9. Continuity of linear function is characterised by
(a) Hahn-Banach theorem
(b) The open mapping theorem
(c) The closed graph theorem
(d) The principle of uniform bounded ness
10. The legal regulations of parallelogram is nor satisfied by

Earning:   Approval pending.