On critical cardinalities related to Q-sets

2013/06/30 by Taras Banakh, Michal Machura, Lubomyr Zdomskyy · 1 citation
Mathematics · #math.LO #math.GN #msc:03E17 #msc:54H05

paper · pdf

published as Math. Bull. Shevchenko Sci. Soc. 11 (2014), 21-32 · 8 pages

arxiv created 2014/06/04 · arxiv updated 2016/02/23

Abstract

In this note we collect some known information and prove new results about the small uncountable cardinal \mathfrak q₀. The cardinal \mathfrak q₀ is defined as the smallest cardinality |A| of a subset A⊂ \mathbb R which is not a Q-set (a subspace A⊂\mathbb R is called a Q-set if each subset B⊂ A is of type F_σ in A). We present a simple proof of a folklore fact that \mathfrak p≤\mathfrak q₀≤min\\mathfrak b,non(\mathcal N),log(\mathfrak c⁺)\, and also establish the consistency of a number of strict inequalities between the cardinal \mathfrak q₀ 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 \mathfrakp < \mathfraklr < \mathfrakq₀, where \mathfraklr denotes the linear refinement number. Another new result is the upper bound \mathfrak q₀\lenon(\mathcal I) holding for any \mathfrak q₀-flexible cccc σ-ideal \mathcal I on \mathbb R.

Cited by

• Completeness and reflexivity type properties of B₁(X)