Partitions of non-complete Baire metric spaces

On Σ^1_1-complete Equivalence Relations on the Generalized Baire Space

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are Σ^1_1-complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in ZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is Σ^1_1-complete (it is, if V = L, but can be forced not to be).

Solutions to congruences using sets with the property of Baire

Hausdorff's paradoxical decomposition of a sphere with countably many points removed (the main precursor of the Banach-Tarski paradox) actually produced a partition of this set into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the set which sends A to B), B is congruent to C, and A is congruent to (B union C). While refining the Banach-Tarski paradox, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the sphere with rotations witnessing the congruences. The purpose of the present paper is to characterize t…

Baire Category for Monotone Sets

We study Baire category for subsets of 2^omega that are downward-closed with respect to the almost-inclusion ordering (on the power set of the natural numbers, identified with 2^omega). We show that it behaves better in this context than for general subsets of 2^omega. In the downward-closed context, the ideal of meager sets is prime and b-complete (where b is the bounding number), while the complementary filter is g-complete (where g is the groupwise density cardinal). We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results for measure …

Completely Baire spaces, Menger spaces, and projective sets

W. Hurewicz proved that analytic Menger sets of reals are $σ$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We al…

The isomorphism relation of theories with S-DOP in generalized Baire spaces

We study the Borel-reducibility of isomorphism relations in the generalized Baire space $κ^κ$. In the main result we show for inaccessible $κ$, that if $T$ is a classifiable theory and $T'$ is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of $T$ is Borel reducible to the isomorphism of models of $T'$. In fact we show the consistency of the following: If $κ$ is inaccessible and $T$ is a superstable theory with S-DOP, then the isomorphism of models of $T$ is $Σ_1^1$-complete.

On Kuratowski partitions

In 1935 K. Kuratowski posed the problem whether a function f:X ! Y , (X is completely metrizable and Y is metrizable), with the property that a preimage of each open has the Baire property, is continuous apart from a meager set. This paper is a selection of older results related to the question posed by Kuratowski in 1935, coming from among others Solovay and Bukovsk?y, and quite new ones concerning considerations in Ellentuck topology and some propertiesof K-ideals, (i.e. ideals associated with Kuratowski partitions).

Bad Wadge-like reducibilities on the Baire space

We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classic…

On large cardinals and generalized Baire spaces

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal $κ$. We show the consistency of $E^{λ^{++},λ^{++}}_{λ\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_λ$ in the space $(λ^{++})^{λ^{++}}$, being continuously reducible to $E^{2,λ^{++}}_{λ^+\text{-club}}$, the relation of equivalence modulo the non-stationary ideal restricted to $S^{λ^{++}}_{λ^+}$ in the space $2^{λ^{++}}$. Then we show the consistency of $E^{2,κ}_{reg}$…

Between Polish and completely Baire

All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3) Every closed subspace of $X$ is either scattered or it contains a homeomorphic copy of $2^ω$. (4) Every closed subspace of $X$ is a Baire space. While (4) is the well-known property of being completely Baire, properties (2) and (3) have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property re…

A groupoid formulation of the Baire Category Theorem

We prove that the Baire Category Theorem is equivalent to the following: Let G be a topological groupoid such that the unit space is a complete metric space, and there is a countable cover of G by neighbourhood bisections. If G is effective, then G is topologically principal.

On partitions of Ellentuck-large sets

It is proved that no non-meager subspace of the space $[ω]^ω$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.

Non-meager free sets and independent families

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leqω$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such that $F$ is $R$-free for every $R\in\mathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $ω$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\le…

广义贝尔空间的开放双超图二分法及其应用

The open dihypergraph dichotomy for generalized Baire spaces and its applications

研究广义贝尔空间的开放双超图二分法，通过两种方式将相关结果提升到${}^\kappa\kappa$，并应用此方法解决了关于广义贝尔空间可定义子集的多个问题，如胡列维茨二分法变体等。

Finite generators for countable group actions in the Borel and Baire category settings

For a continuous action of a countable discrete group $G$ on a Polish space $X$, a countable Borel partition $P$ of $X$ is called a generator if $G \cdot P := \{ gC : g \in G, C \in P \}$ generates the Borel $σ$-algebra of $X$. For $G = Z$, the Kolmogorov--Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite entropy. It was asked by Benjamin Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We…

Topology, Metric Spaces and the Generalized Continuum Hypothesis

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal …

确定性公理下零维齐性空间的完全分类

A complete classification of the zero-dimensional homogeneous spaces under determinacy

本文在确定性公理下对零维齐性空间及ω上滤子给出完全分类，以拓扑复杂性和Baire范畴刻画，并得到半滤子与滤子的纯拓扑刻画及Wadge理论闭包性质结果。

Some remarks on Kuratowski partitions

We introduce the notion of $K$-ideals associated with Kuratowski partitions and we prove that each $κ$-complete ideal on a measurable cardinal $κ$ can be represented as a $K$-ideal. Moreover, we show some results concerning precipitous and Fréchet ideals.

度量空间的秩问题

The rank problem for metric spaces

该研究解决了度量空间的秩问题，构造出具有不可数Scott秩的完备第二可数度量空间，证明Doucha给出的秩上界ω₁是紧的，还开发了基于Baire纲的秩变体并用于给出秩上界的新证明。

在完备离散赋值域中的巴纳赫-塔斯基悖论

The Banach-Tarski paradox in complete discretely valued fields

该研究在完备离散赋值域中探讨了巴纳赫-塔斯基悖论，证明了域及其球体在等距群下的悖论分解，并在局部紧致条件下建立了有界子集的等分解性。