Let M be a subspace of R where \(M= [1,2]\cup [3,4]\) then \([1,2]\) is
Correct Answer: |
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C) both open and closed in M |
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Description for Correct answer:
Clearly [1, 2] is closed in M = [1, 3]\(\cup\) [3, 4]
\(\textbf{Result:}\)
Let M be a metric space and \(M_{1}\) is a subspace of M.
Let \(A_{1}\subseteq M_{1}\). Then \(A_{1}\) is open in \(M_{1}\) if and only if there exists an open set A of M that \(A_{1}=A\cup M_{1}\)
Let \(\Large A= \left(\frac{1}{2},\frac{5}{2}\right) \)
and let \(A_{1}=[1, 2]\) then clearly
\(\Large A_{1}=[1,2]= \left(\frac{1}{2},\frac{5}{2}\right)\cap M \)
\(\therefore [1,2]\) is open in M.
Therefore \([1,2]\) is both open and closed in M.
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