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 $$\textbf{ZF}$$ ZF a new characterization of iso-dense spaces in terms of special quasiorders. For a non-empty family $$\mathcal {A}$$ A of subsets of a set X, a quasiorder $${{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}$$ ≲ A on X determined by $$\mathcal {A}$$ A is defined. Necessary and sufficient conditions for $$\mathcal {A}$$ A are given to have the property that the topology consisting of all $${{\,\mathrm{\lesssim }\,}}_{\mathcal {A}}$$ ≲ A -increasing sets coincides with the generalized topology on X consisting of the empty set and all supersets of non-empty members of $$\mathcal {A}$$ A . The results obtained, applied to the quasiorder $${{\,\mathrm{\lesssim }\,}}_{\mathcal {D}}$$ ≲ D determined by the family $$\mathcal {D}$$ 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.