# Two small uncountable cardinals related to Q-sets

A subset $$A$$ of the real line is called a Q-set if any subset of of $$A$$ is of type $$F_\sigma$$ in $$A$$.

Let $$\mathfrak q_0$$ be the smallest cardinality of a subset $$X\subset\mathbb R$$ which is not a Q-set in $$\mathbb R$$.

It can be shown that $$q_0$$ is the smallest cardinality of a set $$A\subset \mathbb R$$ containing two disjoint subsets $$A_1,A_2$$ that cannot be separated by $$F_\sigma$$-subsets of $$\mathbb R$$.

We say that two subsets $$A,B\subset\mathbb R$$ can be separated by $$F_\sigma$$-sets if there are two disjoint $$F_\sigma$$-sets $$\tilde A$$ and $$\tilde B$$ in $$\mathbb R$$ such that $$A\subset\tilde A$$ and $$B\subset\tilde B$$.

Taking into account the characterization of $$\mathfrak q_0$$ in terms of $$F_\sigma$$-separation, let us define two modifications of $$\mathfrak q_0$$.

Let $$\mathfrak q_1$$ be the smallest cardinal $$\kappa$$ for which there exists a set $$A\subset \mathbb R$$ of cardinality $$|A|\le\kappa$$ and a family $$\mathcal B$$ of compact subsets of $$\mathbb R$$ such that $$|\mathcal B|\le\kappa$$, $$A\cap\bigcup\mathcal B=\emptyset$$ and the sets $$A$$ and $$\bigcup\mathcal B$$ cannot be separated by $$F_\sigma$$-subset of $$\mathbb R$$.

Let $$\mathfrak q_2$$ be the smallest cardinal $$\kappa$$ for which there exists two families $$\mathcal A$$ and $$\mathcal B$$ of compact subsets of $$\mathbb R$$ such that $$\max\{|\mathcal A|,|\mathcal B|\}\le\kappa$$, the set $$\bigcup\mathcal A$$ and $$\bigcup\mathcal B$$ are disjoint but cannot be separated by $$F_\sigma$$-subsets of $$\mathbb R$$.

It is clear that $$\mathfrak q_2\le \mathfrak q_1\le \mathfrak q_0$$. By analogy with the proof of Theorem 2 in this paper, it can be shown that $$\mathfrak p\le\mathfrak q_2$$. So, we have the inequalities $$\mathfrak p\le\mathfrak q_2\le\mathfrak q_1\le\mathfrak q_0.$$ The cardinal $$\mathfrak q_0$$ has been studied in the literature. What about the cardinals $$\mathfrak q_1$$ and $$\mathfrak q_2$$?

Problem 1. Have the cardinal characteristics $$\mathfrak q_1$$ and $$\mathfrak q_2$$ been considered in the literature?

Problem 2. Are the strict inequalities $$\mathfrak q_2<\mathfrak q_1$$ and/or $$\mathfrak q_1<\mathfrak q_0$$ consistent?