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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 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 {\, \, } ₀ ≲ 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 {\, \, } ₀ ≲ 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 {\, \, } ₃ ≲ 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.
Richmond et al. (Wed,) studied this question.