On critical cardinalities related to $Q$-sets

Abstract

In this note we collect some known information and prove new results about the small uncountable cardinal ${\mathfrak{q}}_0$. The cardinal ${\mathfrak{q}}_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb{R}$ which is not a $Q$-set (a subspace $A\subset \mathbb{R}$ is called a $Q$-set if each subset $B\subset A$ is of type $F_{\sigma }$ in $A$). We present a simple proof of a folklore fact that $\mathfrak{p}\le {\mathfrak{q}}_0\le \min \{\mathfrak{b},\mathrm{non}(\mathcal{N}),\log ({\mathfrak{c}}^{+})\}$, and also establish the consistency of a number of strict inequalities between the cardinal ${\mathfrak{q}}_0$ and other standard small uncountable cardinals. This is done by combining some known forcing results. A new result of the paper is the consistency of $\mathfrak{p}<\mathfrak{lr}<{\mathfrak{q}}_0$, where $\mathfrak{lr}$ denotes the linear refinement number. Another new result is the upper bound ${\mathfrak{q}}_0\le \mathrm{non}(\mathcal{I})$ holding for any ${\mathfrak{q}}_0$-flexible cccc $\sigma$-ideal $\mathcal{I}$ on $\mathbb{R}$.

8 pages

References in corpus (1)

• Hereditarily supercompact spaces

Cited by in corpus (4)

• Verbal covering properties of topological spaces
• First-countable Lindelöf scattered spaces
• $\sigma$-Continuous functions and related cardinal characteristics of the continuum
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