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