-spaces, perfect spaces and related cardinal characteristics of the continuum
Volume 125 / 2023
Banach Center Publications 125 (2023), 9-15
MSC: Primary 54D10; Secondary 03E15, 03E17, 03E35, 03E50, 54A35, 54H05.
DOI: 10.4064/bc125-1
Abstract
A topological space X is called a Q-space if every subset of X is of type F_\sigma in X. For i\in\{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 q_1\le\mathfrak q_2\le\mathfrak q_3. For i\in\{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 q_3 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\in\{1,2,3\}.
