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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?
Verified 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.Most Upvoted 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.
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