2021/01/08 by Kyriakos Keremedis, Keremedis, Kyriakos, Eleftherios Tachtsis +3
Computer Science · Mathematics · #03E25 #03E35 #54D35 #54E35 #54G12 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2101.02825
openalex publication_date 2021/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In ZF, in the absence of the axiom of choice, basic properties of iso-dense spaces are investigated. A new permutation model is constructed in which a discrete weakly Dedekind-finite space can have the Cantor set as a remainder. 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 countable axiom of choice for families of finite sets. Results concerning the problem of whether it is provable in ZF that every non-discrete compact metrizable space contains an infinite compact scattered subspace are also included.