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Roll No. ........................
(ii) If X is a Hausdorff space, then every
convergent sequence in X has :
convergent sequence in X has :
Total No. of Questions : 11
[Total No. of Printed Pages : 8
KL - 485
M. A. / M. Sc. Mathematics (Sem. II)
Regular/Private/ATKT/EX
Examination - June - 2023
Examination - June - 2023
MATHEMATICS
Paper III
Topology-II
Paper III
Topology-II
Time : 3 Hours
Maximum Marks : 85
Note : Attempt all questions.
Section A
Objective Type Questions 1ΒΌΓ10=15
Objective Type Questions 1ΒΌΓ10=15
1.
Choose the correct answer :
(i) Any finite Tβ-space is :
(ii) If X is a Hausdorff space, then every
convergent sequence in X has :
convergent sequence in X has :
(iii) Every compact subspace of the real line
is :
is :
(iv) If X = {a, b, c} and T = {Ο, {a},
{a, b}, X} is a topology on X, then X is :
{a, b}, X} is a topology on X, then X is :
(v) The product space X Γ Y is compact if
and only if :
and only if :
(vi) Every projection map is :
(vii) If (X, T) be any indiscrete topological
space, then every net in X converges to :
space, then every net in X converges to :
(viii) Number of filters on the empty set
Ο is :
Ο is :
(ix) The relation β and p are......................................
(x) The fundamental group of the circle
is........................................................
is........................................................
Section B
Short Answer Type Questions 5Γ5=25
Short Answer Type Questions 5Γ5=25
2.
Define Tβ-space. Prove that a topological space
is a Tβ-space iff each point is a closed set.
is a Tβ-space iff each point is a closed set.
Or
Define Hausdorff space. Prove that a one-to-one
continuous mapping of a compact space
onto a Hausdorff space is a homeomorphism.
continuous mapping of a compact space
onto a Hausdorff space is a homeomorphism.
3.
Define compact space. Prove that any closed
subspace of a compact space is compact.
subspace of a compact space is compact.
Or
Define continuous mapping. Show that any
continuous image of a compact space is
compact.
continuous image of a compact space is
compact.
4.
Define standard subbase for product topology.
Also define product topology.
Also define product topology.
Or
Prove that product of any non-empty class of
Hausdorff spaces is Hausdorff space.
Hausdorff spaces is Hausdorff space.
5.
Define the following :
(a) Convergence of Nets
(b) Filter
(c) Ultrafilter.
Or
If F be a filter on a set X and S be the
associated net x β X; then show that the F
converges to x as a filter iff S converges to x
as net.
associated net x β X; then show that the F
converges to x as a filter iff S converges to x
as net.
Section C
Long Answer Type Questions 5Γ9=45
Long Answer Type Questions 5Γ9=45
6.
Define the following :
(a) Path Homotopy
(b) Loop
(c) Fundamental Group
(d) Covering Space
Or
Show that the map p : R β SΒΉ given by the
equation :
p(x) = (cos 2Οx, sin 2Οx)
is a covering map.
equation :
p(x) = (cos 2Οx, sin 2Οx)
is a covering map.
7.
If X be a normed space, F a closed subspace
and f a continuous real function defined on F,
whose values lie in [a, b], then show that f has
a continuous extension f' defined on all of X
whose values also lie in [a, b].
and f a continuous real function defined on F,
whose values lie in [a, b], then show that f has
a continuous extension f' defined on all of X
whose values also lie in [a, b].
Or
Prove that the product of spaces is locally
connected iff each co-ordinate space is locally
connected and all except finitely many of them
are connected.
connected iff each co-ordinate space is locally
connected and all except finitely many of them
are connected.
8.
Prove that for a topological space X the
following statement are equivalent :
following statement are equivalent :
(a) X is normal.
(b) For any closed C and any open set G
containing C, there exists an open set H
such that C β H and H β G.
(c) For any closed C and any open set G
containing C, there exists an open set H
and a closed set K such that C β K β H β G.
(b) For any closed C and any open set G
containing C, there exists an open set H
such that C β H and H β G.
(c) For any closed C and any open set G
containing C, there exists an open set H
and a closed set K such that C β K β H β G.
Or
Prove that every closed and bounded subspace
of the real line is compact.
of the real line is compact.
Or
Prove that in a sequentially compact
metric space, every open cover has a Lebesgue
number.
metric space, every open cover has a Lebesgue
number.
9.
Prove that the product of any non-empty class
of compact spaces is compact.
of compact spaces is compact.
10.
Prove that a topological space is compact iff
every ultrafilter in it is convergent.
every ultrafilter in it is convergent.
Or
If S : D β X be a net in a topological space
and x β X, then prove that x is a cluster point
of S iff there exists a subnet of S which
converges to x in X.
and x β X, then prove that x is a cluster point
of S iff there exists a subnet of S which
converges to x in X.
11.
If p: E β B be a covering map p(eβ) = bβ.
The map F: I Γ I β B be continuous with
F(0, 0) = bβ. Then there is a lifting of F to a
continuous map FΜ: I Γ I β E
such that FΜ(0, 0) = eβ.
If F is path homotopy, show that FΜ is also path
homotopy.
The map F: I Γ I β B be continuous with
F(0, 0) = bβ. Then there is a lifting of F to a
continuous map FΜ: I Γ I β E
such that FΜ(0, 0) = eβ.
If F is path homotopy, show that FΜ is also path
homotopy.
Or
Show that a polynomial equation :
xβΏ + aβxβΏβ»ΒΉ + ....... + aα΅£x + aβ = 0
of degree n > 0 with real or complex
co-efficients has at least one (real or complex)
root.
xβΏ + aβxβΏβ»ΒΉ + ....... + aα΅£x + aβ = 0
of degree n > 0 with real or complex
co-efficients has at least one (real or complex)
root.