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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not?
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Consider R² with the usual topology and A,B subset or equal R² then ch...
- Introduction:
In this problem, we are asked to determine whether the interior of the union of two subsets in R² is equal to the union of their interiors.
- Explanation:
- Interior of A union B:
The interior of a set A is the set of all interior points of A. For a point to be an interior point of A, there must exist an open ball centered at that point contained entirely within A.
The interior of A union B, denoted as int(A ∪ B), is the set of all points that are interior points of the union of A and B.
- Union of interiors:
The union of the interiors of A and B, denoted as int(A) ∪ int(B), is the set of all points that are interior points of A or B.
- Conclusion:
In general, it is not true that the interior of the union of two sets is equal to the union of their interiors. There are counterexamples that can be constructed where this equality does not hold. Therefore, we cannot always say that int(A ∪ B) = int(A) ∪ int(B) for arbitrary subsets A and B of R².
In this problem, we are asked to determine whether the interior of the union of two subsets in R² is equal to the union of their interiors.
- Explanation:
- Interior of A union B:
The interior of a set A is the set of all interior points of A. For a point to be an interior point of A, there must exist an open ball centered at that point contained entirely within A.
The interior of A union B, denoted as int(A ∪ B), is the set of all points that are interior points of the union of A and B.
- Union of interiors:
The union of the interiors of A and B, denoted as int(A) ∪ int(B), is the set of all points that are interior points of A or B.
- Conclusion:
In general, it is not true that the interior of the union of two sets is equal to the union of their interiors. There are counterexamples that can be constructed where this equality does not hold. Therefore, we cannot always say that int(A ∪ B) = int(A) ∪ int(B) for arbitrary subsets A and B of R².
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not?
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Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not?.
Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider R² with the usual topology and A,B subset or equal R² then check weather interior(A union B)=(interior(A) )union (interior(B)) or not?.
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