Mathematics Exam > Mathematics Questions > Which one of the following is correct?a)The i...
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Which one of the following is correct?
- a)The intersection of any finite number of open sets is open.
- b)The intersection of an arbitrary number of open sets is open.
- c)The union of an arbitrary family of open sets is closed.
- d)The intersection of any finite number of open sets is closed.
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Which one of the following is correct?a)The intersection of any finite...
Let G1, and G2 be two open sets. Then, if G1 ∩ G2 = φ, it is open
if G1 ∩ G2 ≠ φ, let x ε G1 ∩ G2
implies x ε G1, and x ε G2
implies G1, G2 are nbd of x.
implies G1 ∩ G2 is a nbd of x.
but since x is any point o f G1 ∩ G2 therefore G1 ∩ G2 is a nbd o f each of its points. Hence G1 ∩ G
Now, consider the open sets
implies
which is not an Open set.
implies The intersection of arbitrary number of open set need not be open. Next, let G be the union of an arbitrary family
of open sets, A being an index set. To prove that G is an open set, we shall show that for any point x ε G, it contains an open interval containing x.
Let x ε G implies
atleast one member, say 
of F such that x ε 
. Since, 
is an set, 
an open interval ln such that 
Thus, the set G contains an open interval containing any point x of G. Hence, G is an open set.
if G1 ∩ G2 ≠ φ, let x ε G1 ∩ G2
implies x ε G1, and x ε G2
implies G1, G2 are nbd of x.
implies G1 ∩ G2 is a nbd of x.
but since x is any point o f G1 ∩ G2 therefore G1 ∩ G2 is a nbd o f each of its points. Hence G1 ∩ G
2
is open.Now, consider the open sets
implies
which is not an Open set.implies The intersection of arbitrary number of open set need not be open. Next, let G be the union of an arbitrary family
of open sets, A being an index set. To prove that G is an open set, we shall show that for any point x ε G, it contains an open interval containing x.Let x ε G implies
atleast one member, say of F such that x ε . Since, is an set, an open interval ln such that Thus, the set G contains an open interval containing any point x of G. Hence, G is an open set.
Most Upvoted Answer
Which one of the following is correct?a)The intersection of any finite...
Explanation:
Definition: A set is said to be open if every point in the set is an interior point.
Option A states that the intersection of any finite number of open sets is open. Let's prove this statement.
Proof:
Suppose we have a finite collection of open sets: A₁, A₂, ..., Aₙ.
Let x be an arbitrary point in the intersection of these sets, i.e., x ∈ ⋂ᵢ₌₁ⁿ Aᵢ.
Since x is in the intersection, it means that x is in every set Aᵢ.
Claim: x is an interior point of the intersection of these sets.
Proof of Claim:
To show that x is an interior point, we need to find an open ball centered at x that is completely contained in the intersection of the sets.
Since x is in every set Aᵢ, there exists an open ball B(x, rᵢ) centered at x such that B(x, rᵢ) ⊆ Aᵢ for each i.
Let r be the minimum of all the radii rᵢ, i.e., r = min{r₁, r₂, ..., rₙ}.
Now consider the open ball B(x, r). We claim that B(x, r) is completely contained in the intersection of the sets.
Let y be an arbitrary point in B(x, r). We need to show that y is also in every set Aᵢ.
Since y is in the ball B(x, r), the distance between y and x is less than r, i.e., d(y, x) < />
By the triangle inequality, we have:
d(y, x) ≤ d(y, x) + d(x, y) = d(y, x) + d(y, x) < r="" +="" r="" />
Since 2r < rᵢ="" for="" each="" i,="" it="" follows="" that="" d(y,="" x)="" />< rᵢ="" for="" each="" />
Therefore, y is in every set Aᵢ, and hence, y is in the intersection of the sets.
This shows that every point x in the intersection of the finite number of open sets has an open ball centered at x that is completely contained in the intersection. Hence, the intersection is open.
Therefore, option A is correct: The intersection of any finite number of open sets is open.
Definition: A set is said to be open if every point in the set is an interior point.
Option A states that the intersection of any finite number of open sets is open. Let's prove this statement.
Proof:
Suppose we have a finite collection of open sets: A₁, A₂, ..., Aₙ.
Let x be an arbitrary point in the intersection of these sets, i.e., x ∈ ⋂ᵢ₌₁ⁿ Aᵢ.
Since x is in the intersection, it means that x is in every set Aᵢ.
Claim: x is an interior point of the intersection of these sets.
Proof of Claim:
To show that x is an interior point, we need to find an open ball centered at x that is completely contained in the intersection of the sets.
Since x is in every set Aᵢ, there exists an open ball B(x, rᵢ) centered at x such that B(x, rᵢ) ⊆ Aᵢ for each i.
Let r be the minimum of all the radii rᵢ, i.e., r = min{r₁, r₂, ..., rₙ}.
Now consider the open ball B(x, r). We claim that B(x, r) is completely contained in the intersection of the sets.
Let y be an arbitrary point in B(x, r). We need to show that y is also in every set Aᵢ.
Since y is in the ball B(x, r), the distance between y and x is less than r, i.e., d(y, x) < />
By the triangle inequality, we have:
d(y, x) ≤ d(y, x) + d(x, y) = d(y, x) + d(y, x) < r="" +="" r="" />
Since 2r < rᵢ="" for="" each="" i,="" it="" follows="" that="" d(y,="" x)="" />< rᵢ="" for="" each="" />
Therefore, y is in every set Aᵢ, and hence, y is in the intersection of the sets.
This shows that every point x in the intersection of the finite number of open sets has an open ball centered at x that is completely contained in the intersection. Hence, the intersection is open.
Therefore, option A is correct: The intersection of any finite number of open sets is open.
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Which one of the following is correct?a)The intersection of any finite number of open sets is open.b)The intersection of an arbitrary number of open sets is open.c)The union of an arbitrary family of open sets is closed.d)The intersection of any finite number of open sets is closed.Correct answer is option 'A'. Can you explain this answer?
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Which one of the following is correct?a)The intersection of any finite number of open sets is open.b)The intersection of an arbitrary number of open sets is open.c)The union of an arbitrary family of open sets is closed.d)The intersection of any finite number of open sets is closed.Correct answer is option 'A'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which one of the following is correct?a)The intersection of any finite number of open sets is open.b)The intersection of an arbitrary number of open sets is open.c)The union of an arbitrary family of open sets is closed.d)The intersection of any finite number of open sets is closed.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following is correct?a)The intersection of any finite number of open sets is open.b)The intersection of an arbitrary number of open sets is open.c)The union of an arbitrary family of open sets is closed.d)The intersection of any finite number of open sets is closed.Correct answer is option 'A'. Can you explain this answer?.
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