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Topology MCQs - This file contains the MCQs covering the online lectures given during COVID-19
Introduction to topology (MATH3119)
University of Education
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BSF 1702379 Topology(Dr. Ghous)
Lecture 1
- Topology studies geometric properties of objects that remain unchanged under (a)Continuous deformations (b)Discontinuous deformations (c)Abstract deformations (d)None of these
- Which one is not a continuous deformation? (a)Stretching (b)Bending (c)Tearing (d)Twisting
- 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.
- 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)
- 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)
- The space (ℝ푛,푑) is called __________. (a)Real metric space (b)n-dimensional Euclidean space (c)n-dimensional real space (d)None of these.
Lecture 2
- 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
- The space C[a, b] for any functions f, g is a metric space under the metric defined by _______
(a)푑(푓,푔)=∫푎푏|푓(푥)−푔(푥)|푑푥 (b)푑(푓,푔)= sup푥∈[푎,푏]sup푥 ∈[푎,푏]|푓(푥)−푔(푥)|
(c)푑(푓,푔)= inf푥∈[푎,푏]|푓(푥)−푔(푥)| (d)Both a and b
BSF 1702379 Topology(Dr. Ghous)
- The set X makes a ______ metric space under the metric defined by 푑(푥,푦)={1 푖푓 푥 ≠ 푦0 푖푓 푥 = 푦
(a)Discrete (b)Bounded (c)Continuous (d)Both a and b
- The metric 푑(푥,푦)={1 푖푓 푥 ≠ 푦0 푖푓 푥 = 푦 defining a discrete metric space (X, d) is called _______.
(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
The set 퐵(푥 0 ;푟)={푥 ∈ 푋:푑(푥,푥 0 )< 푟} with center x 0 and radius r is called (a)Open Ball (b)Close Ball (c)Open Circle (d)Close Circle
The set 퐵(푥 0 ;푟)={푥 ∈ 푋:푑(푥,푥 0 )≤ 푟} with center x 0 and radius r is called (a)Open Ball (b)Close Ball (c)Open Circle (d)Close Circle
The set S(푥 0 ;푟)={푥 ∈ 푋:푑(푥,푥 0 )= 푟} with center x 0 and radius r is called (a)Ball (b)Sphere (c)Open Sphere (d)Close Sphere
For x, y ∈ ℝ푛, the metric defined by 푑 0 (푥,푦)= sup|푥푖−푦푖| is called ______ on ℝ푛. (a)Euclidean Metric (b)Product Metric (c)Postman Metric (d)Usual Metric
For x, y ∈ ℝ푛, the metric defined by 푑 1 (푥,푦)=∑ |푛푖=1푥푖−푦푖| is called ______ on ℝ푛. (a)Euclidean Metric (b)Product Metric (c)Postman Metric (d)Usual Metric
For x, y ∈ ℝ푛, the metric defined by 푑(푥,푦)=√∑ (푖=1푛 푥푖−푦푖) 2 is called ______ on ℝ푛. (a)Euclidean Metric (b)Product Metric
BSF 1702379 Topology(Dr. Ghous)
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
The largest open set of any set is its _______. (a)Interior (b)Sphere (c)Union of all open sets (d)Both a and c
Which one is not valid in general for the interiors of A and B given by A 0 and B 0 in (X, d)? (a)A ⊆ B ➔ A 0 ⊆ B 0 (b)A 0 ∩ B 0 = (A ∩ B) 0 (c)A 0 ∪ B 0 = (A ∪ B) 0 (d)A 0 ∪ B 0 ⊆ (A ∪ B) 0
Lecture 6
- Topology is a hybrid word composed of two words i. Topos _____ and logy ______. (a)Latin; Greek (b)Greek; Latin (c)Arabic; Greek (d)Greek; French
- From topological point of view, square and circle are ______. (a)Different (b)Same (c)Irrelated (d)None of these
- In topology, we mainly care about the following (a)Arrangement of shapes (b)Deformations between shapes (c)Measurements (d)Both a and b
- Another word for topology is ________. (a)Geometry (b)Geometry of shapes (c)Geometry of position (d)Reverse Geometry
- Topology is more effective in _______. (a)Qualitative study (b)Quantitative study (c)Both a and b (d)None of these
- 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
BSF 1702379 Topology(Dr. Ghous)
- The elements of τ are called ______. (a)Subsets (b)Sub-classes (c)τ -open sets (d)None of these
- Mark the false statement (a)⋃{U ∈ τ | U ∈ ∅} = ∅ (b)⋃{U ∈ τ | U ∈ ∅} = X (c)⋂{U ∈ τ | U ∈ ∅} = X (d)∅, X ∈ τ
Lecture 7
- 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
- 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
- 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)∅
- 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
- Let X=ℝ and the class τ contains ∅, ℝ and all open intervals of the form Aq = (-∞, 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
- 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
- The topology of a set equal to the power set of the set is called ________. (a)Discrete topology (b)Indiscrete topology
BSF 1702379 Topology(Dr. Ghous)
(c)Occasionally open (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
- The intersection of the topologies on a set is _______. (a)A topology (b)Not a topology (c)Occasionally a topology (d)None of these
- The union of the topologies on a set is _________. (a)A topology (b)Not a topology (c)Occasionally a topology (d)None of these
- Let a, b ∈ ℝ with usual order relation i. a < b then the open interval from a to b is (a)(a, b) = {x| a< x< b} ⊂ ℝ (b)(a, b) = {x| a≤ x< b} ⊂ ℝ (c)(a, b) = {x| a< x≤ b} ⊂ ℝ (d)None of these
- A subset A of ℝ is open iff ∀ a ∈ A ,∃ an open interval Ia such that (a)a ∈ Ia ⊂ A (b)a ∈ Ia ⊃ A (c)a ∉ Ia ⊂ A (d)None of these
- The set of all open sets of ℝ i. τu is called ________. (a)Usual topology on ℝ (b)Open topology on ℝ (c)Open component of ℝ (d)Both a and b
BSF 1702379 Topology(Dr. Ghous)
Lecture 11
Which one is the representation of a real plane? (a)ℝ × ℝ (b)ℝ 2 (c)ℝ of ℝ (d)Both a and b
Which one is representation of an open disk D(x, y) of radius r centered at origin in ℝ 2? (a)D={(x, y)| x 2 +y 2 < r 2 } (b)D={(x, y)| x 2 +y 2 = r 2 } (c)D={(x, y)| x 2 +y 2 ≤ r 2 } (d)D={(x, y)| x 2 +y 2 > r 2 }
A subset U of ℝ 2 is open iff ∀ a=(x, y) ∈ U, ∃ an open disk Da such that (a)a ∈ Da ⊆ U (b)a ∈ Da ⊂ U (c)a ∉ Da ⊆ U (d)Both a and b
The set of all open sets of ℝ 2 i. τu is called ________. (a)Usual topology on ℝ 2 (b)Open topology on ℝ 2 (c)Open component of ℝ 2 (d)Both a and b
Which one represents an open n-ball centered at x in ℝ푛?
(a)푩풙=൛풚:√∑ (풊=ퟏ풏 풙풊−풚풊)ퟐ< 풓ൟ (b)퐵푥=൛푦:√∑ (푖=1푛 푥푖−푦푖) 2 ≤ 푟ൟ
(c)퐵푥=൛푦:√∑ (푖=1푛 푥푖+푦푖) 2 < 푟ൟ (d)퐵푥=൛푦:√∑ (푖=1푛 푥푖+푦푖) 2 ≤ 푟ൟ 6. A subset U of ℝn is open iff for every a ∈ U, ∃ an open n-ball Ba such that (a)a ∈ Ba ⊆ U (b)a ∈ Ba ⊂ U (c)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)τ 1 ⊄ τ 2 and τ 2 ⊄ τ 2 (d)Both a and b 8. Which comparison of topologies is false? (a)τ ⊆ τD (b)τInD ⊆ τ (c)Both a and b (d)None of these 9. The collection T = {τi} of all topologies on X is partially ordered by ________. (a)Class exclusion (b)Class Inclusion
BSF 1702379 Topology(Dr. Ghous)
(a)7 (b)1, 3 (c)1, 7 (d)3, 7
- Let X = ℝ with usual topology and 퐵 ={푛 1 |푛 ∈ ℕ∗} , then the only limit point of set B is ____.
(a)0 (b)∞ (c)-∞ (d)
Lecture 14
Derived set of A ______ subset of A. (a)Is always (b)Is never (c)May or maybe not (d)None of these
Derived set of empty set is ______. (a)Empty (b)Not empty (c)Maybe empty (d)Real space
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. A’ is _____. (a){a, b, c, d, e} (b){b, d, e} (c){a, b, d, e} (d){b, c}
Let X = ℝ with usual topology and 퐵 ={푛 1 |푛 ∈ ℕ∗} , then the derived set B’ = ____.
(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)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
BSF 1702379 Topology(Dr. Ghous)
Lecture 15
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
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
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
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
Consider ℝ with usual topology, then a subset 퐴 = {1, 12 , 13 , 14 ,...} is _______.
(a)Closed (b)Open (c)Empty (d)Both a and c
Lecture 16
- A set A is a subset of a set B i. A ⊂ B iff _____. (a)Bc⊂ Ac (b)Ac⊂ Bc (c)Ac ∉ Bc (d)None of these
- (푈푥∩퐴)= ∅ implies that (a)푼풙⊂ 푨푪 (b)푈푥⊂ 퐴 (c)푈푥∉ 퐴퐶 (d)푈푥= 퐴
- A subset A of a topological space X is open iff ∀ x ∈ A ∃ Ux (open set containing x) such that (a)푈푥⊂ 퐴퐶 (b)푼풙⊂ 푨 (c)푈푥∉ 퐴퐶 (d)푈푥= 퐴
BSF 1702379 Topology(Dr. Ghous)
(c)[0, 2] (d)(0, 2] 7. Let A̅ be the closure of A such that A ⊂ X (X being a topological space), then (a)A̅ is closed subset of X. (b)A̅ is closed iff A̅ = A
(c)(A̅̅ഥ̅̅)̅= A̅ (d)All of these 8. For A, B ⊂ X, if A ⊂ B then (a)A̅ ⊂ B̅ (b)A̅ ⊃ B̅ (c)A̅ ⊆ B (d)A ⊇ B̅ 9. Let X be a topological space and A, B ⊂ X, then (a)(푨∪푩)̅̅̅̅̅̅̅̅̅̅= A̅ ∪ B̅ (b)(A∪B)̅̅̅̅̅̅̅̅̅̅= A̅ ∩ B̅ (c)(A ∪B)̅̅̅̅̅̅̅̅̅̅ ∉ A̅ ∪ B̅ (d)(A∪B)̅̅̅̅̅̅̅̅̅̅= (A∩B)̅̅̅̅̅̅̅̅̅̅ 10. A subset A of a topological space X is said to be dense iff _____. (a)A̅ = X (b)A̅ = ∅ (c)A ∪ A’ = X (d)Both a and c 11. Let X = ℝ with usual topology and consider ℚ ⊂ ℝ then (a)ℚ is dense in ℝ (b)ℚഥ = ℝ (c)Both a and b (d)None of these
Lecture 18
- Consider ℝ with usual topology and consider the sequence { 1 푛}={1, 12 , 13 ,...} then
(a)Limit pt. of { 1 푛}= 0 (b){ 1 푛}→ 0
(c)Both a and b (d)None of these
- Let X be an indiscrete space and {xn} ⊂ X be a sequence. If a point y ∈ X, then (a)X contains all terms of {xn} (b)Only X contains y (c)푥푛→ 푦 (d)All of these
- In a discrete space X any sequence in X converges to (a)0 (b) (c)Any point in X (d)None of these
BSF 1702379 Topology(Dr. Ghous)
- Let (ℝ, d) be the usual topology and {xn} = {1, 1, 1, ... } in ℝ, then (a){푥푛}→ 1 (b){푥푛} is divergent (c){푥푛} is a constant sequence (d)Both a and c.
- Let (ℝ, d) be the usual topology and {xn} = {1, 2, 3, 4, ... } in ℝ, then (a){푥푛}→ 1 (b){풙풏} 퐢퐬 퐝퐢퐯퐞퐫퐠퐞퐧퐭 (c){푥푛} is a constant sequence (d)Both a and c.
Lecture 19
Let A ⊂ (X, τ). A point a ∈ A is an interior point of A iff ∃ Ua(open set containing a) such that (a)푼풂⊂ 푨 (b)푈푎= 퐴 (c)푈푎∉ 퐴 (d)None of these
Let X = {a, b, c, d} and τ = {∅, {a}, {a, b}, {a, c, d}, X}. Consider A = {a, c} then (a)a is an interior point of A (b)c is an interior point of A (c)a, c, are interior points of A (d)None of these
Let X = ℝ with usual topology and A = [0, 1) then ____ are the interior points of A. (a)0, 0 (b)0, 1 (c)0, 1 (d)[0, 1]
Let (X, τ) be a topological space and A ⊂ X. Then interior of A i. A° = ________. (a){푎 ∈ 퐴|∃ 푈푎∈ 휏,푎 ∈ 푈푎⊂ 퐴} (b){푎 ∈ 퐴|∃ 푈푎∈ 휏,푎 ∈ 푈푎∉ 퐴} (c)Set of all interior points of A (d)Both a and c
Let X = {a, b, c, d} and τ = {∅, {a}, {a, b}, {a, c, d}, X}. Consider A = {a, b, c} then A° = ____. (a){a, b} (b){a, b, c} (c){a, b, d} (d){a, c}
Consider ℝ 2 with usual topology and A = {(x, y) |x = y} then A° = ______. (a)∅ (b)A (c)AC (d)None of these
Let A ⊂ X, then A is open in X iff (a)A = A° (b)A ⊂ A° (c)A ⊃ A° (d)AC = A°
BSF 1702379 Topology(Dr. Ghous)
(a)Basis for τ in X (b)Boundary of X (c)Interior of X (d)Closure of X 2. Let β be a basis for τ on X, then every superclass β* of open subsets of X is ______. (a)Basis for τ in X (b)Boundary of X (c)Interior of X (d)Closure of X 3. Is β = {{a}, {b}, {c, d}} a basis for τ = {∅, {a}, {b}, {a, b}, X}? (a)Yes (b)No (c)Partially (d)Undecided 4. Consider indiscrete space i. a non-empty set X with τ = {∅, X}. then the basis for τ is β = ____. (a){X} (b)∅ (c){XC} (d)None of these 5. The basis β of a set X generates a discrete topology on X, if it contains all elements as ______. (a)Singletons (b)Sets with 2 elements (c)Sets with 3 elements (d)The set X 6. Consider usual topology 휏푢 on ℝ then usual basis 훽푢 for 휏푢 is _______. (a)set of all open intervals (b)set of all closed intervals (c)set of all singletons (d){X} 7. Consider usual topology 휏푢 on ℝ 2 , then usual basis 훽푢 for 휏푢 is _______. (a)Set of all open disks (b)Set of all open triangles (c)Set of all open squares (d)All of these 8. Consider usual topology 휏푢 on ℝ푛, then usual basis 훽푢 for 휏푢 is _______. (a)Set of all n-balls (b)Set of all spheres (c)Set of all pentagons (d)All of these
Lecture 22
The basis Bℓ for lower limit topology on ℝ i. ℝℓ is given by the set ______. (a){[a, b) | a, b ∈ ℝ } (b){(a, b] | a, b ∈ ℝ } (c){[a, b] | a, b ∈ ℝ } (d){(a, b) | a, b ∈ ℝ } ∪ {(a, b)\K | a, b ∈ ℝ }
The basis BUP for upper limit topology on ℝ i. ℝUP is given by the set ______.
BSF 1702379 Topology(Dr. Ghous)
(a){[a, b) | a, b ∈ ℝ } (b){(a, b] | a, b ∈ ℝ } (c){[a, b] | a, b ∈ ℝ } (d){(a, b) | a, b ∈ ℝ } ∪ {(a, b)\K | a, b ∈ ℝ }
- Let K = {1, 12 , 13 ,...} then the basis Bk for K-topology on ℝ i. ℝk is given by the set ______.
(a){[a, b) | a, b ∈ ℝ } (b){(a, b] | a, b ∈ ℝ } (c){[a, b] | a, b ∈ ℝ } (d){(a, b) | a, b ∈ ℝ } ∪ {(a, b)\K | a, b ∈ ℝ } 4. Let Τ퐴={푉 = 푈 ∩퐴|푈 ∈ Τ} be subspace topology of A relative to (X, Τ), then basis BA = ______. (a){푩푨= 푩∩푨|푩 ∈ 휷} (b){퐵퐴= 퐵 ∪퐴|퐵 ∈ 훽} (c){퐵퐴⊂ 퐵 ∩퐴|퐵 ∈ 훽} (d)Both a and c 5. If β = {{a} , {b} , {a, c, d} } is basis for Τ on (X, Τ) then for the subset A={a, b, c} , BA = ________. (a) {{a} , {b} , {a, c, d} } (b) {{a} , {b} , {a, c} } (c) {{a} , {b} , {a, b} , {a, c, d} } (d)None of these
Lecture 23
- For (X, Τ) with basis β, a sub-collection S⊂ Τ is called a sub-basis for Τ iff ∀B ∈ β each B can be written as ________ of elements of S. (a)Closed subsets (b)Open subsets (c)Arbitrary union (d)Finite intersection
- Any class A of subsets of a nonempty set X is ________ for a unique topology Τ on X. (a)Basis (b)Interior (c)Derived Set (d)Exterior
- β = {∅, {n} ,[푛,푛 +1] ,ℝ| n∈ ℝ } is a basis for _________ on ℝ. (a)Discrete topology (b)Indiscrete topology (c)Subspace topology (d)All of these
- Let 풮 be the sub-basis for τ, then a sub-basis for τA on A(where A⊂ X) i. 풮A = ________. (a){S ∩ A|S ∈ 풮} (b){S ∪ A|S ∈ 풮} (c){S ∩ A|S ⊃ 풮} (d){S ∩ A|S = 풮}
- If τ 1 and τ 2 are 2 topologies on a set X, then the union τ 1 ∪ τ 2 is _______. (a)Always a topology (b)May or maybe not a topology (c)Basis for τ 3 on X (d)Both b and c
Topology MCQs - This file contains the MCQs covering the online lectures given during COVID-19
Course: Introduction to topology (MATH3119)
University: University of Education
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