FMS_MCQ_QUESTION_BANK.pdf

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SRM INSTITUTE OF SCIENCE AND TECHNOLOGY
S.R.M NAGAR, KATTANGULATHUR-1603 203
FORM T-2DEPARTMENT OF COMPUTATIONAL INTELLIGENCE
18AIC209T Foundation of Metrics Space
II Year IV Semester
MCQ - QUESTION BANK

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Topic 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.
Topic 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
(a)𝑑(𝑓, 𝑔) = ∫
𝑏
|𝑓(𝑥) − 𝑔(𝑥)| 𝑑𝑥 (b)𝑑(𝑓, 𝑔) = sup sup𝑥 ∈ [𝑎, 𝑏] ฀|𝑓(𝑥) − 𝑔(𝑥)|฀
𝑎
(c)𝑑(𝑓, 𝑔) = inf
𝑥∈[𝑎,𝑏]
|𝑓(𝑥) − 𝑔(𝑥)| (d)Both a and b
𝑥∈[𝑎,𝑏]
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𝑖=1
3. The set X makes a metric space under the metric defined by 𝑑(𝑥, 𝑦) =
1 𝑖𝑓 𝑥 ≠ 𝑦
{
0 𝑖𝑓 𝑥 = 𝑦
(a)Discrete (b)Bounded
(c)Continuous (d)Both a and b
4. The metric 𝑑(𝑥, 𝑦) =
1 𝑖𝑓 𝑥 ≠ 𝑦
{
0 𝑖𝑓 𝑥 = 𝑦
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)
Topic 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) = ∑𝑛 |𝑥𝑖 − 𝑦𝑖| is called on ℝ𝑛.
For x, y ∈ ℝ𝑛, the metric defined by 𝑑(𝑥, 𝑦) = √∑𝑛 (𝑥𝑖 − 𝑦𝑖)2 is called on ℝ𝑛.
𝑖=1
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Topic 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, 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
Topic 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(x; r) ⊆ A (b)x ∉ B(x; r) ⊆ A
(c)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
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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
𝖴 B0
= (A 𝖴 B)0
(d)A0
𝖴 B0
⊆ (A 𝖴 B)0
Topic 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
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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 ∈ τ | U ∈ ∅} = X
(c)⋂{U ∈ τ | U ∈ ∅} = X (d)∅, X ∈ τ
Topic 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 Aq = (-∞, q) where q ∈ ℚ ,
then τ is
(a)Always a topology (b)Not a topology
Topic 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 .
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3. Which one is the smallest possible topology on a set?
4. Which one is the largest possible topology on a set?
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 .
8. A set X with indiscrete topology is called .
Topic 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 (b)Always closed
(c)Occasionally open (d)Occasionally closed
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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
Topic 10
1. The intersection of the topologies on a set is .
(a)A topology (b)Not a topology
2. The union of the topologies on a set is .
(a)A topology (b)Not a topology
3. Let a, b ∈ ℝ with usual order relation i.e. 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
4. 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
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
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Topic 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)D={(x, 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 (b)a ∈ Da ⊂ U
(c)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 (b)Open topology on ℝ2
(c)Open component of ℝ2 (d)Both a and b
5. Which one represents an open n-ball centered at x in ℝ𝑛 ?
(a)𝑩𝒙 = ൛𝒚: √∑𝒏 (𝒙𝒊 − 𝒚𝒊)𝟐 < 𝒓ൟ (b)𝐵𝑥 = ൛𝑦: √∑𝑛 (𝑥𝑖 − 𝑦𝑖)2
≤ 𝑟ൟ
𝒊=𝟏 𝑖=1
(c)𝐵𝑥 = ൛𝑦: √∑𝑛 (𝑥𝑖 + 𝑦𝑖)2
< 𝑟ൟ (d)𝐵𝑥 = ൛𝑦: √∑𝑛 (𝑥𝑖 + 𝑦𝑖)2
≤ 𝑟ൟ
𝑖=1 𝑖=1
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)τ1 ⊄ τ2 and τ2 ⊄ τ2 (d)Both a and b
8. Which comparison of topologies is false?
(a)τ ⊆ τD (b)τInD ⊆ τ
9. The collection T = {τi} of all topologies on X is partially ordered by .
(a)Class exclusion (b)Class Inclusion
(c)Inclusion-Exclusion Principle (d)Both a and c
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Topic 12
1. For A ⊂ X in (X, τ), the collection 𝑟𝐴 = {𝑉 = 𝑈 ∩ 𝐴|𝑈 𝜖 𝑟} 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. 𝑟𝑍 is on ℤ relative to usual topology 𝑟𝑅.
(c)Subspace topology (d)Both a and c
3. Let (𝐴, 𝑟𝐴) be a subspace of (𝑋, 𝑟), 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 𝖴 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 ∈ 𝑟𝑅 such that
(a){x} = U ∩ ℤ (b){x} = U 𝖴 ℤ
(c){x} ⊂ U ∩ ℤ (d){x} ⊆ U ∩ ℤ
Topic 13
1. Let (X, τ) be a topological space and A⊂ X, then x ∈ X is a limit pt. of A, iff ∀𝑈𝑥 ∈ 𝑟, 𝑥 ∈ 𝑈𝑥, we’ve
(a)A ∩ (Ux{x}) ≠ ∅ (b)A ∩ (Ux{x}) = ∅
(c)A 𝖴 (Ux{x}) ≠ ∅ (d)A 𝖴 (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.
(a)7 (b)1.5, 3
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5. Let X = ℝ with usual topology and 𝐵 = {
1
|
𝑛
𝑛 ∈ ℕ∗} , then the only limit point of set B is .
(a)0 (b)∞
(c)-∞ (d)1
Topic 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}
4. Let X = ℝ with usual topology and 𝐵 = {
1
|
𝑛
𝑛 ∈ 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
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Topic 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
5. Consider ℝ with usual topology, then a subset 𝐴 = {1,
1
2
,
1
,
1
3 4
, …} is .
(a)Closed (b)Open
(c)Empty (d)Both a and c
Topic 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 (b)𝑈𝑥 ⊂ 𝐴
(c)𝑈𝑥 ∉ 𝐴𝐶 (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 (b)𝑼𝒙 ⊂ 𝑨
(c)𝑈𝑥 ∉ 𝐴𝐶 (d)𝑈𝑥 = 𝐴
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4. Let X = {a, b, c, d}, τ = {∅, {a}, {a, c},{a, b, d},X} then mark the open set of the following:
(a)A = { a, b, d} (b)B = {c, d}
(c)C = {a, d} (d)D = {a, c, d}
5. Consider ℝ2 with usual topology and 𝐴 = {(𝑥, 𝑦) ∈ ℝ2|𝑥 = 𝑦}. Then the set A is .
(a)Open (b)Not open
6. Consider ℝ2 with usual topology and B= {(𝑥, 𝑦) ∈ ℝ2|1 < 𝑥2 + 𝑦2 < 4}. Then the set B is .
(a)Open (b)Not open
Topic 17
1. If the class of all closed subsets of X containing A is ✔𝐴, then the closure of A denoted as A̅ is .
(a)𝑨
ഥ = ⋂𝑪∈➀𝑨
𝑪
(c)𝐴 ∈ ⋂𝐶∈✔𝐴
𝐶
(b)𝐴 = ⋃𝐶∈✔𝘗
𝐶
(d)None of these
2. Let C = {X, {c, d, e},{a, b, e}, {e}, {a}, ∅} be closed collection of X = {a, b, c, d, e}, and A = {c}
then A
̅ = .
(a){c, d, e} (b){a, b, e}
(c)X (d)∅
3. Let A̅ be the closure of A, then
(a)A ⊂ A̅ (b)A̅ is smallest closed superset of A
(c)A ⊃ A̅ (d)Both a and b
4. The closure of an empty set is .
(a)Always empty (b)Maybe empty
(c)Not empty (d)X
5. If A is a subset of a topological space X, A̅ is closure and A’ is derived set of A, then
(a)A̅ = A 𝖴 A’ (b)A̅ = A ∩ A’
(c)A̅ 𝖴 A = A’ (d)A̅ ∩ A = A’
6. For X = ℝ with usual topology and A = (0, 2], the closure of A i.e. A
̅ = .
(a){0, 2} (b)(0, 2)
(c)[0, 2] (d)(0, 2]
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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)(A
ഥ
) = 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)(
𝑨
𝖴
𝑩
) = A̅ 𝖴 B̅ (b)(
A
𝖴
B
) = A̅ ∩ B̅
(c)(A
𝖴
B
) ∉ A𝖴 B (d)(
A
𝖴
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 𝖴 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)ℚ
ഥ = ℝ
Topic 18
1. Consider ℝ with usual topology and consider the sequence {
1
} = {1,
1
,
1
, … } then
𝑛 2 3
(a)Limit pt. of {
1
𝑛
} = 0 (b){
1
𝑛
} → 0
2. 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)𝑥𝑛 → 𝑦 (d)All of these
3. In a discrete space X any sequence in X converges to
(a)0 (b)1
(c)Any point in X (d)None of these
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4. Let (ℝ, d) be the usual topology and {xn} = {1, 1, 1, … } in ℝ, then
(a){𝑥𝑛} → 1 (b){𝑥𝑛} is divergent
(c){𝑥𝑛} is a constant sequence (d)Both a and c.
5. Let (ℝ, d) be the usual topology and {xn} = {1, 2, 3, 4, … } in ℝ, then
(a){𝑥𝑛} → 1 (b){𝒙𝒏} 𝐢𝐬 𝐝𝐢𝐯𝐞𝐫𝐠𝐞𝐧𝐭
(c){𝑥𝑛} is a constant sequence (d)Both a and c.
Topic 19
1. Let A ⊂ (X, τ). A point a ∈ A is an interior point of A iff ∃ Ua(open set containing a) such that
(a)𝑼𝒂 ⊂ 𝑨 (b)𝑈𝑎 = 𝐴
(c)𝑈𝑎 ∉ 𝐴 (d)None of these
2. 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
3. Let X = ℝ with usual topology and A = [0, 1) then are the interior points of A.
(a)0.2, 0.3 (b)0, 1
(c)0, 1.1 (d)[0, 1]
4. Let (X, τ) be a topological space and A ⊂ X. Then interior of A i.e. A° = .
(a){𝑎 ∈ 𝐴|∃ 𝑈𝑎 ∈ 𝑟, 𝑎 ∈ 𝑈𝑎 ⊂ 𝐴} (b){𝑎 ∈ 𝐴|∃ 𝑈𝑎 ∈ 𝑟, 𝑎 ∈ 𝑈𝑎 ∉ 𝐴}
(c)Set of all interior points of A (d)Both a and c
5. 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}
6. Consider ℝ2 with usual topology and A = {(x, y) |x = y} then A° = .
(a)∅ (b)A
(c)AC
(d)None of these
7. Let A ⊂ X, then A is open in X iff
(a)A = A° (b)A ⊂ A°
(c)A ⊃ A° (d)AC
= A°
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𝑪
𝐶
Topic 20
1. Let A ⊂ (X, τ). A point a ∈ AC
is an exterior point of A iff ∃ Ua(open set containing a) such that
(a)𝑼𝒂 ⊂ 𝑨𝑪 (b)𝑈𝑎 = 𝐴𝐶
(c)𝑈𝑎 ∉ 𝐴𝐶 (d)None of these
2. If A ⊂ (X, τ) and AC
be its complement then interior of AC
is denoted by .
(a)Ext(A) (b)Int(AC
)
(c)(AC
)° (d)All of these
3. Let (X, τ) be a topological space and A ⊂ X. Then exterior of A i.e. (AC
)° = .
(a){𝑎 ∈ 𝐴𝐶|∃ 𝑈𝑎 ∈ 𝑟, 𝑎 ∈ 𝑈𝑎 ⊂ 𝐴𝐶} (b){𝑎 ∈ 𝐴𝐶|∃ 𝑈𝑎 ∈ 𝑟, 𝑎 ∈ 𝑈𝑎 ∉ 𝐴𝐶}
(c)Set of all exterior points of A (d)Both a and c
4. Let X = ℝ with usual topology and ℚ ⊂ ℝ then Ext (ℚ) = .
(a)ℚ (b)ℚ’
(c)∅ (d)ℝ
5. Let (X, τ) be a topological space and A⊂ X then a point x ∈ X is a boundary pt. of A iff ∀ Ux, we’ve
(a)𝑼𝒙 ∩ 𝑨 ≠ ∅ 𝒂𝒏𝒅 𝑼𝒙 𝖴 𝑨 ≠ ∅ (b) 𝑈𝑥 ∩ 𝐴 ≠ ∅ 𝑎𝑛𝑑 𝑈𝑥 𝖴 𝐴 = ∅
(c) 𝑈𝑥 ∩ 𝐴 = ∅ 𝑎𝑛𝑑 𝑈𝑥 𝖴 𝐴 ≠ ∅ (d) 𝑈𝑥 ∩ 𝐴 = ∅ 𝑎𝑛𝑑 𝑈𝑥 𝖴 𝐴 = ∅
6. Another name for boundary point is .
(a)frontier point (b)continuity point
(c)divergence point (d)Both a and b
7. The boundary of a set A ⊂ X denoted by bd(A) contains all boundary points of A such that
(a)𝒃𝒅(𝑨) = ൫𝑨° 𝖴 𝑬𝒙𝒕(𝑨)൯ (b) 𝑏𝑑(𝐴) = ൫𝐴° 𝖴 𝐸𝑥𝑡(𝐴)൯
(c) 𝑏𝑑(𝐴) = ൫𝐴° ∩ 𝐸𝑥𝑡(𝐴)൯ (d) 𝑏𝑑(𝐴) = ൫𝐴° ∩ 𝐸𝑥𝑡(𝐴)൯
8. The boundary of a set A is empty iff A is .
(a)Open (b)Close
(c)Empty (d)Both open and close
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Topic 21
1. Every element of τ can be written as a union of elements of β if β is
(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.e. 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 𝑟𝑢 on ℝ then usual basis 𝛽𝑢 for 𝑟𝑢 is .
(a)set of all open intervals (b)set of all closed intervals
(c)set of all singletons (d){X}
7. Consider usual topology 𝑟𝑢 on ℝ2, then usual basis 𝛽𝑢 for 𝑟𝑢 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 𝑟𝑢 on ℝ𝑛, then usual basis 𝛽𝑢 for 𝑟𝑢 is .
(a)Set of all n-balls (b)Set of all spheres
(c)Set of all pentagons (d)All of these
Topic 22
1. The basis Bℓ for lower limit topology on ℝ i.e. ℝ𝘗 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 } 𝖴 {(a, b)K | a, b ∈ ℝ }
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2.The basis BUP for upper limit topology on ℝ i.e. ℝUP is given by the set .
(a){[a, b) | a, b ∈ ℝ } (b){(a, b] | a, b ∈ ℝ }
3. Let K = {1,
1
,
1
2 3
, … } then the basis Bk for K-topology on ℝ i.e. ℝk is given by the set .
(a){[a, b) | a, b ∈ ℝ } (b){(a, b] | a, b ∈ ℝ }
4. Let Τ𝐴 = {𝑉 = 𝑈 ∩ 𝐴|𝑈 ∈ Τ} be subspace topology of A relative to (X, Τ), then basis BA = .
(a){𝑩𝑨 = 𝑩 ∩ 𝑨|𝑩 ∈ 𝖰} (b){𝐵𝐴 = 𝐵 𝖴 𝐴|𝐵 ∈ 𝛽}
(c){𝐵𝐴 ⊂ 𝐵 ∩ 𝐴|𝐵 ∈ 𝛽} (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
Topic 23
1. 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
2. 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
3. β = {∅, {n} , [𝑛, 𝑛 + 1] ,ℝ| n∈ ℝ } is a basis for on ℝ.
(c)Subspace topology (d)All of these
4. Let 𝗌 be the sub-basis for τ, then a sub-basis for τA on A(where A⊂ X) i.e. 𝗌A = .
(a){S ∩ A|S ∈ 𝗌} (b){S 𝖴 A|S ∈ 𝗌}
(c){S ∩ A|S ⊃ 𝗌} (d){S ∩ A|S = 𝗌}
5. If τ1 and τ2 are 2 topologies on a set X, then the union τ1 𝖴 τ2 is .
(a)Always a topology (b)May or maybe not a topology
(c)Basis for τ3 on X (d)Both b and c

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