Topology-I - BU 2022 Question Paper

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IJ-443

M.A./M.Sc. 1^st Semester (Reg./Pvt./ATKT)

Examination, 2022-23

Maths

Paper - III

Topology-I

Time : 3 Hours

[Maximum Marks : Reg.= 85

Pvt.= 100]

Note :- Attempt all the questions.

SECTION - 'A'

Note :- 5 questions of 3 mark each

Choose the correct answer:

(i) A set is said to be countable if it is :

(a) non-empty and finite
(b) Countably infinite
(c) (a) or (b)
(d) None of the above

(ii) A subset A of a topological space X is said to be dense if:

(a) ¯A = X
(b) ¯∅ = A
(c) ¯X = A
(d) None of the above

(iii) ¯A ∪ ¯B is equal to :

(a) ¯A ∪ ¯B
(b) ¯A ∪ B
(c) ¯A ∪ ¯B
(d) None of the above

(iv) Define separable metric space.

(a) Countable
(b) Connected
(c) Dis connected
(d) None of the above

(v) Every path-connected space is

(a) Countable
(b) Connected
(c) Dis connected
(d) None of the above

SECTION - 'B'

Short Answer Type Questions

5×5=25

Note :- Five question of 5 mark each.

Write down the statements of the following.

(a) Cantors theorem

(b) Zorn's lemma

OR
Prove that a countable union of countable sets is countable.

Let A be an arbitrary subset of the topological space X, then prove that

¯A = {x : each neighbourhood of x intersects A}

OR
Let A be an arbitrary subset of the topological space X, then prove that

(i) ¯A = A ∪ D(A)

(ii) A is closed ⇔ A ⊇ D(A)

Define closure of a set in a topological space and write down the kuratowski closure axioms. OR
Prove that a subset of a topological space is open if and only if it is a neighbourhood of each of its points.

Prove that a countable product of second countable spaces is second countable. OR
Define first countable and second countable space.

SECTION - 'C'

Long Answer Type Questions

9×5=45

Note :- Five questions of 9 mark each.

Define a path and path connected space. OR
Define locally connected and locally path connected space.

Let X and Y be two sets each of which is numerically equivalent to a subset of the other then prove that all of X is numerically equivalent to all of Y. OR
Prove that the finite product of countable sets is countable.

Let T₁ and T₂ be two topologies on a non-empty set X then show that T₁ ∩ T₂ is also a topology on X. OR
Let (X, J) be a topological space then define the following :

(a) Closed Sets

(b) Closure

(d) Neighbourhoods

(e) Interior

(f) Exterior

(g) Boundary

Let X be a set and suppose for each x ∈ X, a non-empty family N_x of subsets of X is give satisfying the following properties:

(i) If U ∈ N_x then x ∈ U
(ii) for any U, V ∈ N_x and U ∩ V ∈ N_x
(iii) if V ∈ N_x and U ⊃ V, then U ∈ N_x
(iv) If U ∈ N_x, then there exists V ∈ N_x such that V ⊂ U and V ∈ N_y for all y ∈ V then prove that there is a unique topology J on X such that for each x ∈ X, N_x coincides with the family of all neighbourhoods of x with respect to J.

OR
Let X be a non-empty set and let there be given a "closure" operation which assigns to each subset A of X a subset ¯A of X such that

(1) ¯φ = φ
(2) A ⊆ ¯A
(3) ¯¯A = ¯A
(4) ¯A ∪ ¯B = ¯(A ∪ B)

If a "closed" set is defined to be one for which ¯A = A then the class of all complements of such sets is a topology on X whose closure operation is precisely that initially given.

10.

Let X be a second countable space. If a non-empty open set G in X is represented as the union of a class {G_j} of open sets then prove that G can be represented as a countable union of G_j's. OR
Prove that every separable metric space is second countable.

11.

Prove that the union of a collection of connected spaces of X that have a point in common is connected. OR
Prove that the image of a connected space under a continuous map is connected.