Mathematics Exam > Mathematics Questions > Which of the following statement is not true?...
Start Learning for Free
Which of the following statement is not true?
- a)finite union of open sets is open
- b)finite intersection of open sets is open
- c)arbitrary union of open sets is open
- d)arbitrary intersection of open sets is open
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
Which of the following statement is not true?a)finite union of open se...
Let us consider An = 
then An is open for each n ∈ ℕ. But
is not open.Free Test
FREE
| Start Free Test |
Community Answer
Which of the following statement is not true?a)finite union of open se...
Arbitrary Intersection of Open Sets is Open
There is a common misconception that the arbitrary intersection of open sets is always open, but this statement is actually not true. It is important to understand the properties of open sets in topology to see why this statement is false.
Explanation:
Finite Union of Open Sets is Open:
- This statement is true because the union of any finite number of open sets is also an open set. This is a fundamental property of open sets in topology.
Finite Intersection of Open Sets is Open:
- This statement is true as well. The intersection of any finite number of open sets is also an open set. This is another important property of open sets.
Arbitrary Union of Open Sets is Open:
- This statement is true. The union of any collection of open sets, whether finite or infinite, is always an open set. This is a key property of open sets in topology.
Arbitrary Intersection of Open Sets is Open:
- This statement is not true. In general, the intersection of an arbitrary collection of open sets may not be open. This can be seen in examples such as the set of real numbers where the intersection of all open intervals containing 0 is the singleton set {0}, which is not open.
In conclusion, while the arbitrary intersection of open sets is not always open, the other statements regarding finite unions, finite intersections, and arbitrary unions of open sets being open hold true in general topology.
There is a common misconception that the arbitrary intersection of open sets is always open, but this statement is actually not true. It is important to understand the properties of open sets in topology to see why this statement is false.
Explanation:
Finite Union of Open Sets is Open:
- This statement is true because the union of any finite number of open sets is also an open set. This is a fundamental property of open sets in topology.
Finite Intersection of Open Sets is Open:
- This statement is true as well. The intersection of any finite number of open sets is also an open set. This is another important property of open sets.
Arbitrary Union of Open Sets is Open:
- This statement is true. The union of any collection of open sets, whether finite or infinite, is always an open set. This is a key property of open sets in topology.
Arbitrary Intersection of Open Sets is Open:
- This statement is not true. In general, the intersection of an arbitrary collection of open sets may not be open. This can be seen in examples such as the set of real numbers where the intersection of all open intervals containing 0 is the singleton set {0}, which is not open.
In conclusion, while the arbitrary intersection of open sets is not always open, the other statements regarding finite unions, finite intersections, and arbitrary unions of open sets being open hold true in general topology.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer?
Question Description
Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. 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 of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. 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 of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer?.
Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. 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 of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. 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 of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Which of the following statement is not true?a)finite union of open sets is openb)finite intersection of open sets is openc)arbitrary union of open sets is opend)arbitrary intersection of open sets is openCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup