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[Total No. of Printed Pages : 8]
Total No. of Questions : 11
GH-485
M.A./M.Sc. IInd Semester (Reg./Pvt./ATKT)
Examination, 2022
Mathematics
Paper - III
Topology-II
Time : 3 Hours]
[Maximum Marks : 85/100
Note :- Attempt the questions from all three sections as directed
SECTION - 'A'
Objective Type Questions
1½×10=15
Note :- Attempt all questions. Each question carries 1½ mark.
1.
Choose the correct answer :
(i)
A topological space X is said to be a T4 - space if it is :
(a) Reguar and T1
(b) Normal and T1
(c) Regular and normal
(d) T0 and T1
(ii)
All metric space are
(a) T0 and T2
(b) T0 and T3
(c) T0, T2, T3 and T4
(d) T0 and T1
(iii)
Every close and bounded subspace of the real line is compact is known as :
(a) The Heine Borel theorem
(b) Tychonoff's theorem
(c) Local compactness
(d) None of the above
(iv)
Every compact metric space is :
(a) Closed compact
(b) Locally compact
(c) Continous
(d) None of the above
(v)
All projection maps are :
(a) Continous maps
(b) Open maps
(c) Both open and continous maps
(d) None of the above
(vi)
The product of any non-empty class of Hausdorff spaces is a :
(a) Compact space
(b) Hausdorff space
(c) Connected space
(d) None of the above
(vii)
Define a directed set and net.
(viii)
Define a filter on a set X.
(ix)
Define path homotopy
(x)
Define homotopy of two maps.
SECTION - 'B'
Short Answer Type Questions
5×5=25
Note :- Attempt all five questions. Each question carry 5 mark.
2.
Prove that every compact subspace of a Hausdorff space is closed.
OR
Define the following :
(i) Hausdorff space
(ii) Normal space
(iii) Regular space
(iv) Completely Regular space
3.
Prove that any continous image of a compact space is compact.
OR
Prove that a topological space is compact if every basic open cover has a finite subcover.
4.
Define product topology and defining open sub-base and defining open base for the product topology.
OR
Prove that the product of any non-empty class of Hausdorff spaces is a Hausdorff space.
5.
Suppose S : D → X is a net and F is a cofinal subset of S. If S | F : F → X convergers to a point x in X, then prove that x is a cluster point of S.
OR
Let B be a family of non-empty subsets of a set X. Then prove that there exists a filter on X having B as a base if and only if B has the property that for any B1, B2 ∈ B, there exists B3 ∈ B such that B1 ∩ B2 ⊃ B3.
6.
Prove that the relation homotopy (=) and path homotopy (≃p) are equivalence relations.
OR
If P : E → B and P' : E' → B' are covering maps, then prove that the map P × P' : E × E' → B × B' is a covering map.
SECTION - 'C'
Long Answer Type Questions
5×9=45
Note :- Attempt all five questions. Each question carry 9 mark.
7.
Prove that every compact Hausdorff space is normal.
OR
Let X be a normal space, and let A and B be disjoint closed subspaces of X. Then prove that there exists a continous real function f defined on X, all of whose values lie in the closed unit interval [0, 1] such that f(A) = 0 and f(B) = 1.
8.
Prove that a topology space is compact if every subbasic open cover has a finite subcover.
OR
Prove the every closed and bounded subspace of the real line is compact.
9.
Prove that the product of any non-empty class of compact spaces is compact.
OR
Prove that product of any non-empty class of connected spaces is connected.
10.
Let S : D → X be a net and F the filter associated with it. Let x ∈ X. Then prove that S converges to x as a net if and ony if F converges to x as a filter. Also x is a cluster point of the net S if and only if x is a cluster point of the filter F.
OR
Prove that a topological space is Hausdorff if and only if no filter can converge to more than one point in its.
11.
State and prove that the function fundamental theorem of algebra.
OR
Let P : E → B be a covering map, p(e0) = b0. Prove that any path f : [0,1] → B beginning at b0 has a unique lifting to a path f in E beginning at e0.