Quasiorders for a Characterization of Iso-dense Spaces

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2024年8月14日

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Tom Richmond,
Eliza Wajch

摘要

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.