Quasiorders for a Characterization of Iso-dense Spaces

Abstract

A (generalized) topological space is called an iso-dense space if the set of all its isolated points is dense in the space. The main aim of the article is to show in \(\textbf{ZF}\) a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family \(\mathcal {A}\) of subsets of a set X, a quasiorder \({{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}\) on X determined by \(\mathcal {A}\) is defined. Necessary and sufficient conditions for \(\mathcal {A}\) are given to have the property that the topology consisting of all \({{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}\)-increasing sets coincides with the generalized topology on X consisting of the empty set and all supersets of non-empty members of \(\mathcal {A}\). The results obtained, applied to the quasiorder \({{\,\mathrm{\lesssim }\,}}_{\mathcal {D}}\) determined by the family \(\mathcal {D}\) of all dense sets of a given (generalized) topological space, lead to a new characterization of non-trivial iso-dense spaces. Independence results concerning resolvable spaces are also obtained.