×

zbMATH — the first resource for mathematics

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.

MSC:
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54E55 Bitopologies
PDF BibTeX XML Cite