2021/08/02 by Kyriakos Keremedis, Keremedis, Kyriakos, Eliza Wajch +1
Economics, Econometrics and Finance · Mathematics · #03E25 #03E35 #54D50 #54D55 #54E50 #Advanced Topology and Set Theory #Economic theories and models #FOS: Mathematics #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2108.01195
openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the absence of the axiom of choice, new results concerning sequential, Fréchet-Urysohn, k-spaces, very k-spaces, Loeb and Cantor completely metrizable spaces are shown. New choice principles are introduced. Among many other theorems, it is proved in ZF that every Loeb, T3-space having a base expressible as a countable union of finite sets is a metrizable second-countable space whose every Fσ-subspace is separable; moreover, every Gδ-subspace of a second-countable, Cantor completely metrizable space is Cantor completely metrizable, Loeb and separable. It is also noticed that Arkhangel'skii's statement that every very k-space is Fréchet-Urysohn is unprovable in ZF but it holds in ZF that every first-countable, regular very k-space whose family of all non-empty compact sets has a choice function is Fréchet-Urysohn. That every second-countable metrizable space is a very k-space is equivalent to the axiom of countable choice for ℝ.