Topology MCQs - This file contains the MCQs covering the online lectures given during COVID-19 PDF

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This file contains the MCQs covering the online lectures given during COVID-19
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Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

Lecture 1 1. Topology studies geometric properties of objects that remain unchanged under (a)Continuous deformations

(b)Discontinuous deformations

(c)Abstract deformations

(d)None of these

2. Which one is not a continuous deformation? (a)Stretching

(b)Bending

(c)Tearing

(d)Twisting

3. Let X be a non-empty set and ℝ be set of real numbers then d: X × X → ℝ is called (a)Metric

(b)Distance function

(c)Metric Space

(d)Both a and b.

4. Which one is incorrect for a distance function d? (a)d(x, y) ≥ 0

(b)d(x, x) = d(y, y)

(c)d(x, y) = d(y, x)

(d)d(x, y) + d(y, z) ≤ d(x, z)

5. For a metric d on a non-empty set X, the metric space is represented as (a)(X, d)

(b)(X, d)→ ℝ

(c)(d, X)

(d)(X:X→ d)

6. The space (ℝ𝑛 , 𝑑) is called __________ . (a)Real metric space

(b)n-dimensional Euclidean space

(c)n-dimensional real space

(d)None of these.

Lecture 2 1. The set C[a, b] of all real continuous functions defined on [a, b] is a subset of the set of all real valued ____ defined on [a, b] (a)Bounded Functions

(b)Unbounded Functions

(c)Discontinuous Functions

(d)None of these

2. The space C[a, b] for any functions f, g is a metric space under the metric defined by _______ 𝑏

(a)𝑑(𝑓, 𝑔) = ∫𝑎 |𝑓(𝑥) − 𝑔(𝑥)| 𝑑𝑥

(b)𝑑(𝑓, 𝑔) = sup sup 𝑥 ∈ [𝑎, 𝑏] ฀|𝑓(𝑥) − 𝑔(𝑥)|฀

(c)𝑑(𝑓, 𝑔) = inf |𝑓(𝑥) − 𝑔(𝑥)|

(d)Both a and b

𝑥∈[𝑎,𝑏]

1|P age

𝑥∈[𝑎,𝑏]

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

1 𝑖𝑓 𝑥 ≠ 𝑦 3. The set X makes a ______ metric space under the metric defined by 𝑑(𝑥, 𝑦) = { 0 𝑖𝑓 𝑥 = 𝑦 (a)Discrete

(b)Bounded

(c)Continuous

(d)Both a and b

1 𝑖𝑓 𝑥 ≠ 𝑦 4. The metric 𝑑(𝑥, 𝑦) = { defining a discrete metric space (X, d) is called _______ . 0 𝑖𝑓 𝑥 = 𝑦 (a)Discrete Metric

(b)Trivial Metric

(c)Instant Metric

(d)Both a and b

5. Which one represents the triangular inequality? (a)d(x, y) + d(y, z) ≤ d(x, z)

(b)d(x, y) + d(y, z) ≥ d(x, z)

(c)d(x, y) + d(y, z) > d(x, z)

(d)d(x, y) + d(y, z) < d(x, z)

Lecture 3 1. The set 𝐵(𝑥0 ; 𝑟) = {𝑥 ∈ 𝑋: 𝑑(𝑥, 𝑥0 ) < 𝑟} with center x0 and radius r is called (a)Open Ball

(b)Close Ball

(c)Open Circle

(d)Close Circle

2. The set 𝐵(𝑥0 ; 𝑟) = {𝑥 ∈ 𝑋: 𝑑(𝑥, 𝑥0 ) ≤ 𝑟} with center x0 and radius r is called (a)Open Ball

(b)Close Ball

(c)Open Circle

(d)Close Circle

3. The set S(𝑥0 ; 𝑟) = {𝑥 ∈ 𝑋: 𝑑(𝑥, 𝑥0 ) = 𝑟} with center x0 and radius r is called (a)Ball

(b)Sphere

(c)Open Sphere

(d)Close Sphere

4. For x, y ∈ ℝ𝑛 , the metric defined by 𝑑0 (𝑥, 𝑦) = sup|𝑥𝑖 − 𝑦𝑖 | is called ______ on ℝ𝑛. (a)Euclidean Metric

(b)Product Metric

(c)Postman Metric

(d)Usual Metric

5. For x, y ∈ ℝ𝑛 , the metric defined by 𝑑1 (𝑥, 𝑦) = ∑ 𝑛𝑖=1|𝑥𝑖 − 𝑦𝑖 | is called ______ on ℝ𝑛. (a)Euclidean Metric

(b)Product Metric

(c)Postman Metric

(d)Usual Metric

6. For x, y ∈ ℝ𝑛 , the metric defined by 𝑑(𝑥, 𝑦) = √∑ 𝑛𝑖=1(𝑥𝑖 − 𝑦𝑖 )2 is called ______ on ℝ𝑛. (a)Euclidean Metric 2|P age

(b)Product Metric

Muhammad ZainUlAbidin Khan BSF1702379

(c)Postman Metric

BS Mathematics 2017-2021 Topology(Dr. Ghous)

(d)Usual Metric

Lecture 4 1. Which one is called a neighborhood of the point x0 ∈ X in a metric space (X, d)? (a)An open ball with center x0

(b)A close ball with center x0

(c)An open set containing x0

(d)Both a and b

2. If for A ⊆ (X X, d), ∃ an open ball B(x; r) ∀ x ∈ A contained in A , then A is called ________ . (a)Open Set

(b)Open Ball

(c)Open Interval

(d)Neighborhood

3. Which one is not an open set? (a)(a, b)

(b)(a, b]

(c)∅

(d)(a, b) U (c, d)

4. If (X, d) is a metric space. Then (a)∅ is open

(b)X is open

(c)X is non-empty

(d)All of these

5. In a discrete space (X, d), every subset is _______ . (a)Open Set

(b)Open Ball

(c)Open Interval

(d)Neighborhood

6. If A represents an open set and A0 represents the interior of A , then (a)A ⊆ A0

(b)A ⊇ A0

(c)A = A0

(d)All of these

Lecture 5 1. A point x of A is an interior point of A if for some r > 0, ∃ an open ball B(x; r) such that (a)x ∈ B B(x; (x; r) ⊆ A

(c)x ∈ B(x; r) ⊇ A

(b)x ∉ B(x; r) ⊆ A (d)None of these

2. For a singleton set A ⊆ ℝ in real line (ℝ, d), and the interior of A denoted by A0, we have (a)A0 = A

(b)A0 = ∅

(c)A0 ⊂ A

(d)None of these

3|P age

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

3. If A is an open interval, then the interior of A is ____ . (a)Equal to interval A.

(b)Empty set (∅)

(c)Subinterval of A

(d)None of these

4. The largest open set of any set is its _______ . (a)Interior

(b)Sphere

(c)Union of all open sets

(d)Both a and c

5. Which one is not valid in general for the interiors of A and B given by A0 and B0 in (X, d)? (a)A ⊆ B ➔ A0 ⊆ B0

(b)A0 ∩ B0 = (A ∩ B)0

(c)A0 ∪ B0 = (A ∪ B)0

(d)A0 ∪ B0 ⊆ (A ∪ B)0

Lecture 6 1. Topology is a hybrid word composed of two words i.e. Topos _____ and logy ______ . (a)Latin; Greek

(b)Greek; Latin

(c)Arabic; Greek

(d)Greek; French

2. From topological point of view, square and circle are ______ . (a)Different

(b)Same

(c)Irrelated

(d)None of these

3. In topology, we mainly care about the following (a)Arrangement of shapes

(b)Deformations between shapes

(c)Measurements

(d)Both a and b

4. Another word for topology is ________ . (a)Geometry

(b)Geometry of shapes

(c)Geometry of position

(d)Reverse Geometry

5. Topology is more effective in _______. (a)Qualitative study

(b)Quantitative study

(c)Both a and b

(d)None of these

6. Which statement is valid for a power set P(X) of a finite set X with n elements? (a)Also referred as a Class

(b)∅ and X ∈ P(X)

(c)|P(X)| = 2n

(d)All of these

4|P age

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

7. The elements of τ are called ______. (a)Subsets

(b)Sub-classes

(c)τ -open sets

(d)None of these

8. Mark the false statement (a)⋃{U ∈ τ | U ∈ ∅} = ∅

(b)⋃{U ⋃{U ∈ τ | U ∈ ∅} = X

(c)⋂{U ∈ τ | U ∈ ∅} = X

(d)∅, X ∈ τ

Lecture 7 1. How many topologies can be made on a 1-point set? (a)Exactly 1

(b)Exactly 2

(c)More than 1

(d)None of these

2. How many topologies can be made on a multiple-points set? (a)Exactly 1

(b)Exactly 2

(c)More than 1

(d)None of these

3. For X = {a, b, c}, τ = {∅, {b}, {a, b}, {b, c}, X} is not a topology because of the absence of _____ . (a)X

(b)Union of some elements

(c)Intersection of some elements

(d)∅

4. Let X=ℝ and the class τ contains ∅, ℝ and all open intervals of the form Ia = (a, ∞), then τ is (a)Always a topology

(b)Not a topology

(c)Occasionally a topology

(d)None of these

5. Let X=ℝ and the class τ contains ∅, ℝ and all open intervals of the form A q = (-∞, q) where q ∈ ℚ , then τ is (a)Always a topology

(b)Not a topology

(c)Occasionally a topology

(d)None of these

Lecture 8 1. The topology of a set containing only ∅ and the set itself is called _______ . (a)Discrete topology

(b)Indiscrete topology

(c)Trivial topology

(d)Both b and c

2. The topology of a set equal to the power set of the set is called ________ . (a)Discrete topology

5|P age

(b)Indiscrete topology

Muhammad ZainUlAbidin Khan BSF1702379

(c)Trivial topology

BS Mathematics 2017-2021 Topology(Dr. Ghous)

(d)Both b and c

3. Which one is the smallest possible topology on a set? (a)Discrete topology

(b)Indiscrete topology

(c)Trivial topology

(d)Both b and c

4. Which one is the largest possible topology on a set? (a)Discrete topology

(b)Indiscrete topology

(c)Trivial topology

(d)Both b and c

5. If X is a set and τ is a topology on X then (X, τ) is called _______ . (a)τ-Topological space

(b)Discrete topological space

(c)Indiscrete topological space

(d)None of these

6. A topological space is _______ on the topology. (a)Dependent

(b)Not dependent

(c)Often not based

(d)None of these

7. A set X with discrete topology is called ________ . (a)τ-Topological space

(b)Discrete topological space

(c)Indiscrete topological space

(d)None of these

8. A set X with indiscrete topology is called ________ . (a)τ-Topological space

(b)Discrete topological space

(c)Indiscrete topological space

(d)None of these

Lecture 9 1. Which intersection is used as the 3rd axiom to satisfy a topology?

(a)Finite intersection

(b)Arbitrary intersection

(c)Infinite intersection

(d)None of these

2. Arbitrary intersection is not used in defining a topology because it ______ to the topology. (a)Does not belong

(b)May or may not belong

(c)Belongs

(d)None of these

3. In a topological space (X, τ), the subclasses ∅ and X are ______. (a)Always open

6|P age

(b)Always closed

Muhammad ZainUlAbidin Khan BSF1702379

(c)Occasionally open

BS Mathematics 2017-2021 Topology(Dr. Ghous)

(d)Occasionally closed

4. Closeness and openness are __________ terms. (a)Relative

(b)Absolute

(c)Topology dependent

(d)Both a and c

5. A set ______ open and close at the same time. (a)Can be

(b)Cannot be

(c)Is always

(d)None of these

6. A subset of a discrete topological space _______ open and close at the same time. (a)Can be

(b)Cannot be

(c)Is always

(d)None of these

7. The collection of all closed subsets A of X does not satisfy the condition (a)∅, X ∈ A

(b) A is closed under arbitrary intersection

(c)A is closed under arbitrary union

(d)None of these

Lecture 10 1. The intersection of the topologies on a set is _______ . (a)A topology

(b)Not a topology

(c)Occasionally a topology

(d)None of these

2. The union of the topologies on a set is _________ . (a)A topology

(b)Not a topology

(c)Occasionally a topology

(d)None of these

3. Let a, b ∈ ℝ with usual order relation i.e. a < b then the open interval from a to b is (a) (a)(a, (a, b) = {x {x|| a< xx< < b} ⊂ ℝ

(c)(a, b) = {x| a< x≤ b} ⊂ ℝ

(b)(a, b) = {x| a≤ x< b} ⊂ ℝ (d)None of these

4. A subset A of ℝ is open iff ∀ a ∈ A ,∃ an open interval Ia such that (a)a ∈ Ia ⊂ A (c)a ∉ Ia ⊂ A

(b)a ∈ Ia ⊃ A (d)None of these

5. The set of all open sets of ℝ i.e. τu is called ________ .

(a)Usual topology on ℝ

(b)Open topology on ℝ

(c)Open component of ℝ

(d)Both a and b

7|P age

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

Lecture 11 1. Which one is the representation of a real plane? (a)ℝ × ℝ

(b)ℝ2

(c)ℝ of ℝ

(d)Both a and b

2. Which one is representation of an open disk D(x, y) of radius r centered at origin in ℝ2 ? (a) (a)D= D= D={(x {(x {(x,, y) y)|| x2+y2 < r2}

(b)D={(x, y)| x2+y2 = r2}

(c)D={(x, y)| x2+y2 ≤ r2}

(d)D={(x, y)| x2+y2 > r2}

3. A subset U of ℝ2 is open iff ∀ a=(x, y) ∈ U, ∃ an open disk Da such that (a)a ∈ Da ⊆ U (c)a ∉ Da ⊆ U

(b) (b)a a ∈ Da ⊂ U

(d)Both a and b

4. The set of all open sets of ℝ2 i.e. τu is called ________ . (a)Usual topology on ℝ2

(c)Open component of ℝ2

(b)Open topology on ℝ2 (d)Both a and b

5. Which one represents an open n-ball centered at x in ℝ𝑛 ? (a)𝑩𝒙 = ൛𝒚: √∑ 𝒏𝒊=𝟏(𝒙𝒊 − 𝒚𝒊 )𝟐 < 𝒓ൟ

(b)𝐵𝑥 = ൛𝑦: √∑ 𝑛𝑖=1(𝑥𝑖 − 𝑦𝑖 )2 ≤ 𝑟ൟ

(c)𝐵𝑥 = ൛𝑦: √∑ 𝑛𝑖=1(𝑥𝑖 + 𝑦𝑖 )2 < 𝑟ൟ

(d)𝐵𝑥 = ൛𝑦: √∑ 𝑛𝑖=1(𝑥𝑖 + 𝑦𝑖 )2 ≤ 𝑟ൟ

6. A subset U of ℝn is open iff for every a ∈ U, ∃ an open n-ball Ba such that (a)a ∈ Ba ⊆ U

(c)a ∉ Ba ⊆ U

(b) (b)a a ∈ Ba ⊂ U

(d)Both a and b

7. Two topologies are comparable iff (a)One is weaker than other

(b)One is finer than other

(c)τ1 ⊄ τ2 and τ2 ⊄ τ2

(d) d)Bot Bot Both h a aand nd b

8. Which comparison of topologies is false? (a)τ ⊆ τD

(b)τInD ⊆ τ

(c)Both a and b

(d) (d)Non Non None eo off tthese hese

9. The collection T = {τi} of all topologies on X is partially ordered by ________ . (a)Class exclusion 8|P age

(b)C (b)Class lass Incl Inclusio usio usion n

Muhammad ZainUlAbidin Khan BSF1702379

(c)Inclusion-Exclusion Principle

BS Mathematics 2017-2021 Topology(Dr. Ghous)

(d)Both a and c

Lecture 12 1. For A ⊂ X in (X, τ), the collection 𝜏𝐴 = {𝑉 = 𝑈 ∩ 𝐴|𝑈 𝜖 𝜏} of subsets of A is called ______ . (a)Subspace topology

(b)Topology on A relative to (X, τ)

(c)Subclass topology

(d)Both a and b

2. 𝜏𝑍 is ________ on ℤ relative to usual topology 𝜏𝑅 . (a)Discrete topology

(b)Indiscrete topology

(c)Subspace topology

(d)Both a and c

3. Let (𝐴, 𝜏𝐴 ) be a subspace of (𝑋, 𝜏), then H⊂ A is relative open to A iff ∃ an open G⊂ X such that (a)H = G ∩ A

(b)H ∩ G = A

(c)H ∩ G ⊆ A

(d)H = G ∪ A

4. Let X = ℝ with usual topology and A = ℤ with relative discrete topology such that A⊂ X then (a){n} ⊂ ℤ are open

(b){n} ⊆ ℤ are open

(c){n} ⊃ ℤ are open

(d){n} - ℤ are open

5. Let X = ℝ with usual topology and A = ℤ then ∀ x ∈ ℤ, ∃ U ∈ 𝜏𝑅 such that (a) (a){x {x {x} = U ∩ ℤ

(c){x} ⊂ U ∩ ℤ

(b){x} = U ∪ ℤ (d){x} ⊆ U ∩ ℤ

Lecture 13 1. Let (X, τ) be a topological space and A⊂ X, then x ∈ X is a limit pt. of A, iff ∀𝑈𝑥 ∈ 𝜏, 𝑥 ∈ 𝑈𝑥 , we’ve

(a) (a)A A ∩ (Ux\{x {x}) ≠ ∅

(c)A ∪ (Ux\{x}) ≠ ∅

(b)A ∩ (Ux\{x}) = ∅

(d)A ∪ (Ux\{x}) = ∅

2. Which name is used for a limit point? (a)Accumulation point

(b)Derived point

(c)Extreme point

(d)Both a and b

3. Let X = {a, b, c, d, e}, τ = {∅, {a}, {c, d}, {a, c, d}, {b, c, d, e},X} and A = {a, b, c} then limit points of A are ______ . (a)a, b, c, d, e

(b)b, d, e

(c)a, b, d, e

(d)b, c

4. Let X = ℝ with usual topology and A = (0, 3) then ____ are the valid limit points of A. 9|P age

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

(a)7

(b)1.5, 3

(c)1.5, 7

(d)3, 7 1

5. Let X = ℝ with usual topology and 𝐵 = { | 𝑛 ∈ ℕ∗ } , then the only limit point of set B is ____. 𝑛 (a)0

(b)∞

(c)-∞

(d)1

Lecture 14 1. Derived set of A ______ subset of A. (a)Is always

(b)Is never

(c)May or maybe not

(d)None of these

2. Derived set of empty set is ______ . (a)Empty

(b)Not empty

(c)Maybe empty

(d)Real space

3. Let X = {a, b, c, d, e}, τ = {∅, {a}, {c, d}, {a, c, d}, {b, c, d, e},X} and A = {a, b, c} then derived set of A i.e. A’ is _____ . (a){a, b, c, d, e}

(b){b, d, e}

(c){a, b, d, e}

(d){b, c} 1

4. Let X = ℝ with usual topology and 𝐵 = { | 𝑛 ∈ ℕ∗ } , then the derived set B’ = ____. 𝑛 (a){0}

(b){∞}

(c){-∞}

(d){1}

5. Let X = ℝ with usual topology and A = ℚ , then the derived set A’ = ____ . (a)ℚ

(b)ℚ’

(c) (c)ℝ ℝ

(d)∅

6. In a discrete space X, the derived set A’ of any subset A is _____. (a)Always Empty

(b)Maybe empty

(c)Always X

(d)Ac

7. In case of indiscrete space X, the derived set A’ of a subset A can be _____ depending on A. (a)A’ = ∅

(b)A’ = Ac

(c)A’ = X

(d)All of these

10 | P a g e

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

Lecture 15 1. A subset A of a topological space X is closed iff ________ . (a)Ac is open

(b)Ac is closed

(c)Ac is empty

(d)None of these

2. A subset A of a topological space X is closed iff ________ . (a)A’ ⊂ A

(b)A’ ∈ A

(c)A’ ⊃ A

(d)Both a and b

3. In a discrete space X, any subset A of X is _____. (a)Closed

(b)May or maybe not closed

(c)Empty

(d)None of these

4. Let X = {a, b, c, d}, τ = {∅, {a}, {a, c},{a, b, d},X} then the subset A = {b, d} is _______ . (a)Closed

(b)Open

(c)Empty

(d)Both a and c 1 1 1

5. Consider ℝ with usual topology, then a subset 𝐴 = {1, 2 , 3 , , …} is _______ . 4 (a)Closed

(b)Open

(c)Empty

(d)Both a and c

Lecture 16 1. A set A is a subset of a set B i.e. A ⊂ B iff _____ . (a)Bc⊂ Ac

(b)Ac⊂ Bc

(c)Ac ∉ Bc

(d)None of these

2. (𝑈𝑥 ∩ 𝐴) = ∅ implies that (a)𝑼𝒙 ⊂ 𝑨𝑪 (c)𝑈𝑥 ∉ 𝐴𝐶

(b)𝑈𝑥 ⊂ 𝐴

(d)𝑈𝑥 = 𝐴

3. A subset A of a topological space X is open iff ∀ x ∈ A ∃ Ux (open set containing x) such that (a)𝑈𝑥 ⊂ 𝐴𝐶 (c)𝑈𝑥 ∉ 𝐴𝐶

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(b)𝑼𝒙 ⊂ 𝑨

(d)𝑈𝑥 = 𝐴

Muhammad ZainUlAbidin Khan BSF1702379

BS Mathematics 2017-2021 Topology(Dr. Ghous)

4. Let X = {a, b, c, d}, τ = {∅, {a}, {a, c},{a, b, d},X} then mark the open set of the fol...