On Iso-dense and Scattered Spaces without $$\textbf{AC}$$

Abstract

A topological space is iso-dense if it has a dense set of isolated points, and it is scattered if each of its non-empty subspaces has an isolated point. In $$\textbf{ZF}$$ ZF (i.e. Zermelo–Fraenkel set theory without the Axiom of Choice ($$\textbf{AC}$$ AC )), basic properties of iso-dense spaces are investigated. A new permutation model is constructed, in which there exists a discrete weakly Dedekind-finite space having the Cantor set as a remainder; the result is transferable to $$\textbf{ZF}$$ ZF . This settles an open problem posed by Keremedis, Tachtsis and Wajch in 2021. A metrization theorem for a class of quasi-metric spaces is deduced. The statement “Every compact scattered metrizable space is separable” and several other statements about metric iso-dense spaces are shown to be equivalent to the axiom of countable choice for families of finite sets. Results related to the open problem of the set-theoretic strength of the statement “Every non-discrete compact metrizable space contains an infinite compact scattered subspace” are also included.