209 research outputs found

    Lebesgue's Density Theorem and definable selectors for ideals

    Full text link
    We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.Comment: 28 pages; minor corrections and a new introductio

    A Special Class of Almost Disjoint Families

    Get PDF
    The collection of branches (maximal linearly ordered sets of nodes) of the tree <ωω{}^{<\omega}\omega (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal -- for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is {\it off-branch} if it is almost disjoint from every branch in the tree; an {\it off-branch family} is an almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal O} is a maximal off-branch family}\}. Results concerning o\frak o include: (in ZFC) ao{\frak a}\leq{\frak o}, and (consistent with ZFC) o\frak o is not equal to any of the standard small cardinal invariants b\frak b, a\frak a, d\frak d, or c=2ω{\frak c}=2^\omega. Most of these consistency results use standard forcing notions -- for example, Con(b=a<o=d=c)Con({\frak b}={\frak a}<{\frak o}={\frak d}={\frak c}) comes from starting with a model of ZFC+CHZFC+CH and adding ω2\omega_2-many Cohen reals. Many interesting open questions remain, though -- for example, Con(o<d)Con({\frak o}<{\frak d})

    Perfect subsets of generalized Baire spaces and long games

    Full text link
    We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space λλ{}^\lambda\lambda, where λ\lambda is an uncountable cardinal with λ<λ=λ\lambda^{<\lambda}=\lambda. In the first main theorem, we show that that the perfect set property for all subsets of λλ{}^{\lambda}\lambda that are definable from elements of λOrd{}^\lambda\mathrm{Ord} is consistent relative to the existence of an inaccessible cardinal above λ\lambda. In the second main theorem, we introduce a Banach-Mazur type game of length λ\lambda and show that the determinacy of this game, for all subsets of λλ{}^\lambda\lambda that are definable from elements of λOrd{}^\lambda\mathrm{Ord} as winning conditions, is consistent relative to the existence of an inaccessible cardinal above λ\lambda. We further obtain some related results about definable functions on λλ{}^\lambda\lambda and consequences of resurrection axioms for definable subsets of λλ{}^\lambda\lambda

    On critical cardinalities related to QQ-sets

    Full text link
    In this note we collect some known information and prove new results about the small uncountable cardinal q0\mathfrak q_0. The cardinal q0\mathfrak q_0 is defined as the smallest cardinality A|A| of a subset ARA\subset \mathbb R which is not a QQ-set (a subspace ARA\subset\mathbb R is called a QQ-set if each subset BAB\subset A is of type FσF_\sigma in AA). We present a simple proof of a folklore fact that pq0min{b,non(N),log(c+)}\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 q0\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 p<lr<q0\mathfrak{p} < \mathfrak{lr} < \mathfrak{q}_0, where lr\mathfrak{lr} denotes the linear refinement number. Another new result is the upper bound q0non(I)\mathfrak q_0\le\mathrm{non}(\mathcal I) holding for any q0\mathfrak q_0-flexible cccc σ\sigma-ideal I\mathcal I on R\mathbb R.Comment: 8 page
    corecore