Mathematics Exam > Mathematics Questions > Every open set of real numbers is the union o...
Start Learning for Free
Every open set of real numbers is the union of
- a)countable collection of disjoint open intervals
- b)uncountable collection of disjoint open intervals
- c)countable collection of disjoint closed intervals.
- d)uncountable collection of disjoint closed intervals
Correct answer is option 'A'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
Every open set of real numbers is the union ofa)countable collection o...
Explanation:
In order to understand why the correct answer is option 'A', let's first define what an open set is and then discuss each option.
Open Set:
An open set in the real numbers is a set that does not contain its boundary points. In other words, for any point in the set, there exists an open interval around that point that is contained entirely within the set.
Option A:
A countable collection of disjoint open intervals can cover every point in an open set. This is because for any point in the open set, there exists an open interval around that point that is contained entirely within the set. By taking the union of all these open intervals, we can cover every point in the open set.
Option B:
An uncountable collection of disjoint open intervals is not necessary to cover every point in an open set. Since the open set contains only real numbers, which are countable, covering every point with an uncountable collection of intervals is not required.
Option C:
A countable collection of disjoint closed intervals cannot cover every point in an open set. This is because a closed interval includes its boundary points, while an open set does not contain its boundary points. So, there will always be points in the open set that are not covered by the closed intervals.
Option D:
An uncountable collection of disjoint closed intervals is not necessary to cover every point in an open set. Similar to option B, the open set contains only countable real numbers, so an uncountable collection of intervals is not required.
Therefore, the correct answer is option 'A' as a countable collection of disjoint open intervals is sufficient to cover every point in an open set.
In order to understand why the correct answer is option 'A', let's first define what an open set is and then discuss each option.
Open Set:
An open set in the real numbers is a set that does not contain its boundary points. In other words, for any point in the set, there exists an open interval around that point that is contained entirely within the set.
Option A:
A countable collection of disjoint open intervals can cover every point in an open set. This is because for any point in the open set, there exists an open interval around that point that is contained entirely within the set. By taking the union of all these open intervals, we can cover every point in the open set.
Option B:
An uncountable collection of disjoint open intervals is not necessary to cover every point in an open set. Since the open set contains only real numbers, which are countable, covering every point with an uncountable collection of intervals is not required.
Option C:
A countable collection of disjoint closed intervals cannot cover every point in an open set. This is because a closed interval includes its boundary points, while an open set does not contain its boundary points. So, there will always be points in the open set that are not covered by the closed intervals.
Option D:
An uncountable collection of disjoint closed intervals is not necessary to cover every point in an open set. Similar to option B, the open set contains only countable real numbers, so an uncountable collection of intervals is not required.
Therefore, the correct answer is option 'A' as a countable collection of disjoint open intervals is sufficient to cover every point in an open set.
Free Test
FREE
| Start Free Test |
Community Answer
Every open set of real numbers is the union ofa)countable collection o...
A is correct
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer?
Question Description
Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect 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 Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect 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 Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer?.
Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect 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 Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect 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 Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer?.
Solutions for Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. 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 Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer?, a detailed solution for Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer? has been provided alongside types of Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Every open set of real numbers is the union ofa)countable collection of disjoint open intervalsb)uncountable collection of disjoint open intervalsc)countable collection of disjoint closed intervals.d)uncountable collection of disjoint closed intervalsCorrect answer is option 'A'. 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