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

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 $i\in \{1,2,3\}$.

arXiv comment6 pages