Define z to be the smallest cardinality of a function f:X->Y with X and Y sets of reals such that there is no Borel function g extending f. In this paper we prove that it is relatively consistent with ZFC to have b<z where b is, as usual, smallest cardinality of an unbounded family in w^w. This answers a question raised by Zapletal. We also show that it is relatively consistent with ZFC that there exists a set of reals X such that the Borel order of X is bounded but there exists a relatively analytic subset of X which is not relatively coanalytic. This answers a question of Mauldin.
Given an analytic equivalence relation, we tend to wonder whether it is Borel. When it is non Borel, there is always the hope it will be Borel on a "large" set -- nonmeager or of positive measure. That has led Kanovei, Sabok and Zapletal to ask whether every proper $σ$ ideal satisfies the following property: given $E$ an analytic equivalence relation with Borel classes, there exists a set $B$ which is Borel and $I$-positive such that $E\restriction_{B}$ is Borel. We propose a related problem -- does every proper $σ$ ideal satisfy: given $A$ an analytic subset of the plane with Borel sections, …
We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of antichains of size continuum, and that there is no finite basis. 2010 Mathematics Subject Classification. 03E15, 54H05
The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Borel equivalence relations on that space. In the case of the Baire space, the two smooth relations are closed and the embedding is homeomorphic.
We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $ω_2$. This implies that $ω_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has length any given limit ordinal less than $ω_2$, e.g., $ω$ or $ω_1+ω_1$. Latex2e: 24 pages plus 8 page appendix Latest version at: www.math.wisc.edu/~miller
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
We modify arguments by Harrington, Marker, and Shelah, 1990, to reprove extensions of two key results in their paper "Borel orderings"in the context of bounding and decomposition of analytic subsets of Borel partial quasi-orderings.
We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bfΣ}^0_1 \!\times\! {\bfΣ}^0_ξ$ sets, or by a ${\bfΠ}^0_1 \!\times\! {\bfΠ}^0_ξ$ set.
The following will be shown: Let $I$ be a $σ$-ideal on a Polish space $X$ with the property that the associated forcing of $I^+$ Borel subsets ordered by $\subseteq$ is a proper forcing. Let E be an analytic or coanalytic equivalence relation on this Polish space with all equivalence classes Borel. If sharps of certain sets exist, then there is an $I^+$ Borel subset $C$ of $X$ such that $E \upharpoonright C$ is a Borel equivalence relation.
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.
We prove that if $\leq$ is an analytic partial order then either $\leq$ can be extended to a (boldface) $Δ^1_2$ linear order similar to an antichain in $2^{<ω_1}$ ordered lexicographically or a certain Borel partial order $\leq_0$ embeds in $\leq.$ Some corollaries for analytic equivalence relations are given, for instance, if $E$ is a $Σ^1_1[z]$ equivalence relation such that $E_0$ does not embed in $E$ then $E$ is determined by intersections with E-invariand Borel sets coded in $L[z]$.
We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an analytic equivalence relation and an analytic quasi-order and we show that their complexity depends only on the order type of $D$. When $D$ contains a decreasing sequence, isometry is Borel bireducible with countable graph isomorphism and isometric embeddability has maximal complexity among analytic quasi-orders. If $D$ is well-ordered the situation is more…
We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (\eg the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Add…
We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean algebra generated by the frame $\mathcal{O}(X)$ of opens to be the "$\infty$-Borel sets" in a locale $X$. We show that several known results in locale theory may be interpreted in this framework as direct analogs of classical descriptive set-theoretic facts, including e.g., the Lusin separation, Lusin-Suslin, and Baire category theorems for locales; we also prove several extensions of these result…
Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L_{ω_1 ω}) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivale…
We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in…
We study the extension of the Kechris-Solecki-Todorcevic dichotomy on analytic graphs to dimensions higher than 2. We prove that the extension is possible in any dimension, finite or infinite. The original proof works in the case of the finite dimension. We first prove that the natural extension does not work in the case of the infinite dimension, for the notion of continuous homomorphism used in the original theorem. Then we solve the problem in the case of the infinite dimension. Finally, we prove that the natural extension works in the case of the infinite dimension, but for the notion of B…
We show that there is a Borel graph on a standard Borel space of Borel chromatic number three that admits a Borel homomorphism to every analytic graph on a standard Borel space of Borel chromatic number at least three. Moreover, we characterize the Borel graphs on standard Borel spaces of vertex-degree at most two with this property, and show that the analogous result for digraphs fails.
Contents: 2. Invited contribution: Ultrafilters and small sets 3. Research announcements 3.1. Inverse Systems and I-Favorable Spaces 3.2. Combinatorial and hybrid principles for sigma-directed families of countable sets modulo finite 3.3. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.4. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension 3.5. Large continuum, oracles 3.6. Borel hierarchies in infinite products of Polish spaces 3.7. A game for the Borel functions 3.8. On some problems in gene…
A theorem of Galvin asserts that if the unordered pairs of reals are partitioned into finitely many Borel classes then there is a perfect set P such that all pairs from P lie in the same class. The generalization to n-tuples for n >= 3 is false. Let us identify the reals with 2^omega ordered by the lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be the least n such that x(n) not= y(n). Let the type of an increasing n-tuple {x_0, ... x_{n-1}}_< be the ordering <^* on {0, ...,n-2} defined by i<^*j iff D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel col…
Omega-powers of finitary languages are omega languages in the form V^omega, where V is a finitary language over a finite alphabet X. Since the set of infinite words over X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers naturally arises and has been raised by Niwinski, by Simonnet, and by Staiger. It has been recently proved that for each integer n > 0, there exist some omega-powers of context free languages which are Pi^0_n-complete Borel sets, and that there exists a context free language L such that L^omega is analytic but not Borel…
A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $μ$ on $G$ such that $μ(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $μ$ such that $μ(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This resul…
We show that the continuum hypothesis implies there exists a Lindelof space X such that X x X is the union of two metrizable subspaces but X is not metrizable. This gives a consistent solution to a problem of Balogh, Gruenhage, and Tkachuk. The main lemma is that assuming the continuum hypothesis there exist disjoint sets of reals X and Y such that X is Borel concentrated on Y, (i.e., for any Borel set B if Y is contained in B then X-B is countable,) but (X x X - diagonal) is relatively Fsigma in (X x X) U (Y x Y).
We study the complexity of the space $C^*_p(X)$ of bounded continuous functions with the topology of pointwise convergence. We are allowed to use descriptive set theoretical methods, since for a separable metrizable space $X$, the measurable space of Borel sets in $C^*_p(X)$ (and also in the space $C_p(X)$ of all continuous functions) is known to be isomorphic to a subspace of a standard Borel space. It was proved by A. Andretta and A. Marcone that if $X$ is a $σ$-compact metrizable space, then the measurable spaces $C_p(X)$ and $C^*_p(X)$ are standard Borel and if $X$ is a metrizable analytic…
We show that every non-Haar-null analytic subset of $\mathbb{Z}^ω$ contains a non-Haar-null closed subset. Moreover, we also prove that the codes of Haar-null analytic subsets, and, consequently, closed Haar-null sets in the Effros Borel space of $\mathbb{Z}^ω$ form a $\mathbfΔ^1_2$ set.
Jayne and Rogers proved that every function from an analytic space into a separable metric space is decomposable into countably many continuous functions with closed domains if and only if the preimage of each $F_σ$ set under it is again $F_σ$. Many researchers conjectured that the Jayne-Rogers theorem can be generalized to all finite levels of Borel functions. In this paper, by using the Shore-Slaman join theorem on the Turing degrees, we show the following variant of the Jayne-Rogers theorem at finite and transfinite levels of the hierarchy of Borel functions: For all countable ordinals $α$ …
We study and compare in two degree-theoretic ways (iterated Halting oracles analogous to Kleene's arithmetical hierarchy and the Borel hierarchy of descriptive set theory) the capabilities and limitations of three models of analytic computation: BSS machines (aka real-RAM) and strongly/weakly analytic machines as introduced by Hotz et. al. (1995).
Omega-powers of finitary languages are languages of infinite words (omega-languages) in the form V^omega, where V is a finitary language over a finite alphabet X. They appear very naturally in the characterizaton of regular or context-free omega-languages. Since the set of infinite words over a finite alphabet X can be equipped with the usual Cantor topology, the question of the topological complexity of omega-powers of finitary languages naturally arises and has been posed by Niwinski (1990), Simonnet (1992) and Staiger (1997). It has been recently proved that for each integer n > 0, there ex…
We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todorcevic dichotomy about analytic gr…
Let $X$ be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel $σ$-additive measure $μ$ which is good in the sense that for any clopen subsets $U,V\subset X$ with $μ(U)<μ(V)$ there is a clopen set $W\subset V$ with $μ(W)=μ(U)$. We study $σ$-ideals with Borel base on $X$ which are invariant under the action of the group $H_μ(X)$ of measure-preserving homeomorphisms of $(X,μ)$, and show that any such $σ$-ideal $\mathcal I$ is equal to one of seven $σ$-ideals: $\{\emptyset\}$, $[X]^{\leω}$, $\mathcal E$, $\mathcal M\cap\mathcal N$, $\mathcal M$, $\mathcal…
