Dense sets and irresolvable spaces. (English) Zbl 0698.54020

Ganster, M.; Reilly, I. L.; Vamanamurthy, M. K.

Ric. Mat. 36, No. 2, 164-170 (1987).

Let D denote the set of all dense subsets of a topological space (X,\({\mathcal T})\), and \(D^ 0\) the set of interiors of the dense subsets. The space is called resolvable if D contains a disjoint pair; hyperconnected if D contains evry nonempty open set. The authors investigate relationship between properties of X and the properties of D and \(D^ 0\). For example, it is proved that \(D^ 0\) is a filter-base if and only if X is irresolvable and that D is an ultrafilter if and only if X is irresolvable and hyperconnected.The authors also prove some properties of D(\({\mathcal T})\) and \(D^ 0({\mathcal T})\), which are the topologies generated by subbases D and \(D^ 0\) respectively. A few sample results are: (i) \({\mathcal T}\) is discrete [resp. resolvable] if and only if D(\({\mathcal T})\) is indiscrete [resp. discrete]. (ii) D(\({\mathcal T})\) has the property that superset of every nonempty open set is open. (iii) D(\({\mathcal T})\) is hyperconnected, if not discrete.

Reviewer: M.G.Murdeshwar

MSC:

54D99

Fairly general properties of topological spaces

Keywords:

resolvable space; hyperconnected space; submaximal space; dense subsets; filter-base; ultrafilter