arXiv subjects
Publications and source records attributed to Jindrich Zapletal.
It is consistent with ZF set theory that the Euclidean topology on the real line is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.
We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$. This shows that a nonprincipal ultrafilter on the integers is not sufficient to construct a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas. This is an improved version of our paper in the Proceedings of the American Mathematical Society, which used a weak version of the Axiom of Choice for the same result.
We study several cardinal characteristics of closed graphs G on compact metrizable spaces. In particular, we address the question when it is consistent for the bounding number to be strictly smaller than the smallest size of a set not covered by countably many compact G-anticliques. We also provide a descriptive set theoretic characterization of the class of analytic graphs with countable coloring number.
Given a Polish space X and a countable family of analytic hypergraphs on X, I consider the sigma-ideal generated by Borel sets which are anticliques in at least one hypergraph in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies. For this broad class of posets, most fusion arguments and iteration preservation arguments can be replaced with simple combinatorial considerations concerning the hypergraphs.
Let M be a transitive model of set theory. There is a canonical interpretation functor between the category of regular Hausdorff, continuous open images of Cech-complete spaces of M and the same category in V, preserving many concepts of topology, functional analysis, and dynamics. The functor can be further canonically extended to the category of Borel subspaces. This greatly simplifies and extends similar results of Fremlin.
In a sigma-closed forcing extension, the bounded forcing axiom for Namba forcing fails. This answers a question of Justin Tatch Moore.
With every $σ$-ideal $I$ on a Polish space we associate the $σ$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $σ$-ideals $I$ and $I^*$ and find connections between their forcing properties. To this end, we associate to a $σ$-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. For $σ$-ideals generated by closed sets we also study the degrees of reals added in the forcing extensions. Among corollaries of our results, we get necessary and sufficient conditions for a $σ$-ideal $I$ generated by closed sets, under which every Borel function can be restricted to an $I$-positive Borel set on which it is either 1-1 or constant.
We isolate a combinatorial property of capacities leading to a construction of proper forcings. Then we show that many classical capacities such as the Newtonian capacity satisfy the property.
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
We study an extensive connection between factor forcings of Borel subsets of Polish spaces modulo a sigma-ideal, and factor forcings of subsets of countable sets modulo an ideal.
If I is a suitably definable sigma-ideal on the real line and the factor algebra of Borel sets modulo I is weakly distributive then the algebra carries a Maharam submeasure.
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
We consider natural cardinal invariants hm_n and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably cov(I)>=hm_n, then non(I) is provably small. The proofs integrate the determinacy theory, forcing and pcf theory.
Many forcing notions obtained using the creature technology are naturally connected with certain integer games.
Let I be a sigma-ideal sigma-generated by a projective collection of closed sets. The forcing with I-positive Borel sets is proper and adds a single real r of an almost minimal degree: if s is a real in V[r] then s is Cohen generic over V or V[s]=V[r].
Whenever x is a tame cardinal invariant and ZFC+large cardinals proves that x=aleph one implies WCG then ZFC+large cardinals proves that x=aleph one implies b=aleph one, and b=aleph one implies WCG. Here WCG is a certain prediction principle on omega one. This theorem is one of the many possible interpolation theorems of this kind.
There is an optimal way to increase certain cardinal invariants of the continuum.
Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.