Properties of some \(*\)-dense-in-itself subsets (Q1777920)

Ideals in a topological space \((X,\tau)\) were studied by \textit{R. Vaidynathaswamy} [Proc. Indian Acad. Sci., Sect. A 20, 51--61 (1944; Zbl 0061.39308)] and \textit{K. Kuratowski} [Topology Vol. I (New York: Academic Press) (1966; Zbl 0158.40802)]. If \(I\) is an ideal in \((X,\tau)\), then for each \(A\subseteq X\), \(\text{cl}^*(A)= A\cup A^*(I,\tau)\), where \(A(I,\tau)= \{x\in X: U\cap A\not\in I\) for each open \(nbd\,U\) of \(x\}\), defines a Kuratowski closure operator, giving a topology \(\tau^*(I,\tau)\), called the \(*\)-topology on \(X\). The concepts of \({\mathcal I}\)-openness and quasi-\({\mathcal I}\)-openness were introduced and studied by \textit{D. Jankovic} and \textit{T. R. Hamlett} [Boll. Unione Mat. Ital., VII. Ser., B 6, No. 3, 453--465 (1992; Zbl 0818.54002)] and \textit{M. E. Abd El-Monsef}, \textit{R. A. Mahmoud} and \textit{A. A. Nasef} [Tamkang J. Math. 31, No. 2, 101--108 (2000; Zbl 0986.54024)]. Again, codense and completely codense ideals were studied by \textit{J. Dontchev} [On pre-\({\mathcal I}\)-open sets and a decomposition of \({\mathcal I}\)-continuity, Banyan Math. J. 2 (1996)], whereas \(*\)-dense in itself and \(*\)-perfect sets were given in [\textit{E. Hayashi}, Math. Ann. 156, 205--215 (1964; Zbl 0129.37702)]. In the present article, the aforesaid notions and certain other ideas are investigated and characterized in terms of the concepts of semiopen, semi-preopen, regular-open, \(\alpha\)-open sets and the like.