Q-spaces, perfect spaces and related cardinal characteristics of the continuum

2022/06/03 by Тарас Банах, Banakh, Taras, Lidiya Bazylevych +1
Computer Science · Mathematics · #03E15 #03E17 #03E35 #03E50 #54A35 #54D10 #54H05 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2206.01667

openalex publication_date 2022/06/03 · openalex created_date 2023/02/12 · openalex updated_date 2026/07/28

Abstract

A topological space X is called a Q-space if every subset of X is of type F_σ in X. For i∈\1,2,3\ let \mathfrak q_i be the smallest cardinality of a second-countable T_i-space which is not a Q-space. It is clear that \mathfrak q₁≤\mathfrak q₂≤\mathfrak q₃. For i∈\1,2\ we prove that \mathfrak q_i is equal to the smallest cardinality of a second-countable T_i-space which is not perfect. Also we prove that \mathfrak q₃ is equal to the smallest cardinality of a submetrizable space, which is not a Q-space. Martin's Axiom implies that \mathfrak q_i=\mathfrak c for all i∈\1,2,3\.

Related

• On symmetrizability and perfectness of second-countable spaces
• S-spaces and large continuum
• k-spaces, sequential spaces and related topics in the absence of the axiom of choice
• Q
• Some Hierarchies of QCB₀-Spaces