Associated topologies of generalized \(\alpha\)-closed sets and \(\alpha\)- generalized closed sets. (English) Zbl 0821.54002

O. Njåstad [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] defined a subset \(A\) of a topological space \((X, \tau)\) to be \(\alpha\)- open if \(A \subset \text{int(cl(int} A))\), and a subset \(B\) of \(X\) to be \(\alpha\)-closed if \(X - B\) is \(\alpha\)-open. The collection \(\tau^ \alpha\) of all \(\alpha\)-open subsets of \((X, \tau)\) is a topology on \(X\), and \(\tau \subset \tau^ \alpha\).
The authors of the paper under review introduce two classes of generalized \(\alpha\)-closed subsets, with the following definitions. A subset \(B\) of \((X, \tau)\) is defined to be \(\alpha\)-generalized closed \([\alpha^{**}\)-generalized closed] in \((X, \tau)\) if \(\tau^ \alpha \text{cl} B \subset U\) \([\tau^ \alpha \text{cl} B \subset \text{int(cl} U)]\) whenever \(B \subset U\) and \(U\) is open in \((X, \tau)\). The paper considers the basic properties of these two classes of subsets, and their associated topologies.


54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54E55 Bitopologies