×

A generalization of the Kuratowski closure-complement problem. (English) Zbl 0891.54005

Summary: By considering the interval \((0,1)\) on the real line it is easy to show that it is not possible, in general, to obtain the boundary of a given set using complementation, closure and interior operations on that set. Therefore one can generalize the Kuratowski closure-complement problem in a special sense. In this paper, we show that if any pair of operations among closure, interior, boundary and complementations may be chosen, then using only two of the operations on any given set \(X\) we obtain that \(X\) belongs to a specific finite family. In addition, for any pair of these operations we give necessary and sufficient conditions that the related family possesses largest cardinal number.

MSC:

54B99 Basic constructions in general topology
PDFBibTeX XMLCite