Andrijevic introduced the notion of b-open set in 1996 as:
A is b-open iff A \subset int(cl(A)) \cup cl(int(A)).
Taras Radul
It is false. A=[0,1)\subset (-1,1)=B. But ClA=[0,1]. A is neither open nor closed in real line with natural topology, B is open, hence b-open.
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Jiarul Hoque
Dear Radul sir, actually, the main question rounded in my mind is not reflected in the question. However, your answer is correct.
My question should be " Does there exist a topological space in which the property cl(A) \subset B for any b-open set B \supset A" hold, where A is neither open nor closed?
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Jiarul Hoque
A is b-open iff A \subset int(cl(A)) \cup cl(int(A)) @S Rajagopal Sridhar
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