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 $\mathbf{ZF}$ 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 ${\lesssim }_{\mathcal{A}}$ 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 ${\lesssim }_{\mathcal{A}}$-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 ${\lesssim }_{\mathcal{D}}$ 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.