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. Princeton University Press , Princeton, N.J. ; University of Tokyo Press (Tokyo, 1974). [5] Etnyre , J. B. 2004 Planar open book decompositions and contact structures

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We introduce the notion of a semi-I-regular set and investigate some of its properties. We show that it is weaker than the notion of a regular-I-closed set. Additionally, we also introduce the notion of an AB I -set by using the semi-I-regular set and study some of its properties. We conclude that a subset A of an ideal topological space (X,τ,I) is open if and only if it is an AB I -set and a pre-I-open set.

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