×

On \(\delta\)-generalized closed sets and \(T_{3/4}\)-spaces. (English) Zbl 0853.54019

If \((X, \tau)\) is a topological space, then the family of regular open subsets of \(X\) is a base for a topology \(\delta\) on \(X\). A subset \(A\) of a topological space \((X, \tau)\) is said to be \(\delta\)-generalized closed (or simply \(\delta\)-g-closed) if \(\text{cl}_\delta (A) \subset U\) whenever \(A\subset U\) and \(U\in \tau\). A space \((X, \tau)\) is a \(T_{3\over 4}\)-space if every \(\delta\)-g-closed subset of \(X\) is closed in \((X, \delta)\). This paper studies the relationship of the \(\delta\)-g-closed sets with other types of closed sets (including generalized-closed, semi-generalized closed, \(\alpha\)-generalized-closed and generalized-semi-preclosed, among others) and the relationship between the separation axiom \(T_{3\over 4}\) and other separation axioms such as \(T_{1\over 2}\) and \(T_1\). The final section of the article concerns \(\delta\)-g-continuous \(\delta\)-g-irresolute functions. A large bibliography is included.

MSC:

54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
PDFBibTeX XMLCite