Quasiorders for a Characterization of Iso-dense Spaces

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