Questions tagged [descriptive-set-theory]

Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.

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About perfect sets and perfect Polish space

Here is the question about the perfect set, If $X$ is a perfect Polish space, and $x\in X$ and $U$ is any open neighborhood of $x$, show that $U\cap X$ is perfect too. “Let’s assume that $U\cap X$ is ...
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Partitions of the real line into Borel subsets

Problem 1. Is it true that for every cardinal $\kappa\le\mathfrak c$ there exists a partition $(B_\alpha)_{\alpha\in\kappa}$ of the real line into $\kappa$ pairwise disjoint non-empty Borel subsets? ...
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Dependent choices (DC) in ${\bf HOD}(\mathbb{R},X)$, where $X$ is a set of reals

In Turing invariant sets and the perfect set property, Math. Log. Quart. 66 (2020), Hamel, Horowitz and Shelah, the authors work in ZF + DC. They claim that DC can be dispensed with, asserting: if $V ...
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A question on Borel equivalence relations

Suppose that $\mathsf E$ is a countable Borel equivalence relation on the reals, and $\mathsf B$ is a finer equivalence of order 2, so that each $\mathsf E$-class consists of precisely two $\mathsf B$-...
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Can we have a “very strong” cone phenomenon in the Turing degrees (and a related question)?

By Borel determinacy + Martin's cone theorem, for every countable fragment $\mathcal{A}$ of $\mathcal{L}_{\omega_1,\omega}$ there is a turing degree ${\bf c}$ such that for all ${\bf d}\ge_T{\bf c}$ ...
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121 views

Scales and concentration

Let $S$ be a dominating subset of $[\mathbb{N}]^{\infty}$. Then $S=\{s_{\alpha} : \alpha<\mathfrak{d}\}$ is called a $\mathfrak{d}$-scale if for every $\alpha<\beta<\mathfrak{d}$, $s_{\beta}\...
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637 views

Descriptive set theory for computer scientists?

It seems to me that there are scattered references of deep relationships between descriptive set theory and computability theory. For one, the relationship between the Borel hierarchy and the ...
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291 views

Proof of the uniform boundedness theorem for analytic subsets of WO

In the paper "A reflection phenomenon in descriptive set theory" by J.P. Burgess, it is stated: Moschovakis [11] has proved the following important Uniform Boundedness Theorem: There is a ...
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56 views

Why measure hyperfinite is equivalent to hyperfinite except for a compressible set?

According to Kechris' paper "The theory of countable Borel equivalence relations" (pp.82), a countable Borel equivalence relation E is measure hyperfinite iff there is an E-invariant Borel ...
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218 views

CH and the density topology on $\mathbb{R}$

In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
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Can there be an upper bound on the Borel rank of the preimages of Borel sets under a surjective Borel map?

Let $X$ and $Y$ be standard Borel spaces, $Y$ uncountable, and $f : X \to Y$ a surjective Borel map. Is it possible that there is a countable ordinal $\alpha$ such that for each Borel set $B \subseteq ...
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183 views

Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?

My general question was is it consistent that any uncountable family of less than $\mathrm{non}(\mathcal{N})$ sets, each with positive measure, has an uncountable subfamily $\mathcal{F}$ s.t. $\bigcap ...
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Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$

Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the ...
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Reference for “$\mathrm{PFA}$ implies $L(\mathbb{R}) \cap \bigcup_{1 \leq k < \omega} \mathcal{P}(\mathbb{R}^k)$ is productive”

The preprint of the recent result of Aspero and Schindler, "Martin's Maximum$^{++}$ implies Woodin's Axiom $(*)$", mentions productive pointclasses, and states that "$\mathrm{PFA}$ ...
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$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?

Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
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What is known about these “explicitly represented” spaces?

Apologies if this is too low-level. A related question that I asked on the Math Stack Exchange got no answers after a year, so I thought it might be better to ask this one here. The standard approach ...
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Non-Borel function from a family of regular Borel measures

Let $X$ and $Y$ be totally disconnected compact Hausdorff spaces. For each $y \in Y$, let $\mu_y$ be a regular Borel probability measure on $X$. Let $\nu$ be a regular Borel probability measure on $Y$....
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Borel complexity of special unions of Polish spaces

Let $X$ be a compact metrizable space and $(A_q)_{q\in\mathbb Q}$ be a family of pairwise disjoint sets, indexed by rational numbers. Assume that the family $(A_q)_{q\in\mathbb Q}$ has the following ...
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How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?

There's a large countable ordinal which has cropped up (as a lower bound!) in a computable structure theory problem I'm playing with. At present I don't really understand how big it is, and I'm ...
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Can two versions of $\omega_1^{CK}(\mathsf{Ord})$ ever coincide?

The goal of this question is to fill in the gap in this old answer of mine. For a transitive set $M$, thought of as an $\{\in\}$-structure, we define the following ordinals (this is not the notation ...
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Is there a Hausdorff space whose “covering problem” has intermediate complexity?

For a "reasonable" pointclass ${\bf \Gamma}$, say that a second-countable space $(X,\tau)$ is ${\bf \Gamma}$-describable iff for some (equivalently, every) enumerated subbase $B=(B_i)_{i\in\...
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Is every element of $\omega_1$ the rank of some Borel set?

It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
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Incidence relations of subspaces with infinite descending flags

Let $W = \prod_{k \in \mathbb N} V_k$ be an infinite product of vector spaces, and let $V = \oplus_{k \in \mathbb N} V_k$ be the corresponding sum. Already the case where $V_k$ is 1-dimensional for ...
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Weakly homogenously Souslin sets and the measurability of $\omega_1$

I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988): The assertion that every set of reals, in $L(\mathbb{R})$, is the ...
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180 views

How to give a Borel set whose projection is not Borel?

I wonder that how Suslin got the idea "analytic set" and the advanced knowledge in this field. Can anyone explain this simply or recommend some reference? All discussions and comments are ...
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275 views

Do Borel subsets of the plane with null sections have Borel projections?

This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping ...
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A rather non-$F_\sigma$ Borel set

I asked this question at MSE a week ago, but received no answer, so I cross-post it here. I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ and ...
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Sierpinski's characterization of $F_{\sigma\delta}$ spaces

According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
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What sets can be unraveled?

A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...
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Is the set of all ICC amenable groups countable?

Is the set of all ICC amenable groups countable? If "yes", then in general, the classes of all countable ICC groups that give rise to the same von Neumann algebra (factor) -- are these ...
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Complexity of the set of models of TA

Recall that the theory of true arithmetic $TA$ is the theory of standard model of arithmetic $\mathcal N$. I am interested in the complexity of the set of models of $TA$ in the lightface or boldface ...
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Comparing generic versions of $\mathbb{R}$

This question was previously asked and bountied at MSE, unsuccessfully. I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...
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219 views

Is the Hilbert cube the countable union of punctiform spaces?

Recall that a (separable) metric space is called punctiform, if all its compact subspaces are zero-dimensional. While "natural" spaces would seem to be punctiform if they already themselves ...
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Trees of prescribed ordinal

My question is very imprecise, as I know very little about descriptive set theory. In a problem I am thinking about I have a family of well-founded trees (finite sequences on $\cup_n X^n$ closed under ...
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Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$

Consider Cantor space $2^\omega$ with the standard topology generated by open sets $[\sigma] = \{ \sigma^\frown x: x \in 2^\omega \}$. If $A \subseteq 2^{<\omega}$ and $x \in 2^\omega$, we say $A$ ...
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Homeomorphisms and “mod finite”

Suppose $f:C\to C$ is a homeomorphism, where $C=\{0,1\}^{\mathbb N}$ is Cantor space. Suppose $f$ preserves $=^*$ (equality on all but finitely many coordinates). Does it follow that $f$ also reflects ...
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What is the Borel complexity of this set?

Problem. What is the Borel complexity of the set $$c(\mathbb Q)=\{(x_n)_{n\in\omega}\in\mathbb R^\omega:\exists\lim_{n\to\infty}x_n\in\mathbb Q\}$$ in the countable product of lines $\mathbb R^\omega$?...
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Non-compact dynamical systems

In topological dynamics, most of the time, we consider the continuous action of a (semi)group $G$ on a compact Hausdorff space $X$. In this context, we can envelop the group in a compact left ...
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Topological proof that a Vitali set is not Borel

This question is purely out of curiosity, and well outside my field — apologies if there is a trivial answer. Recall that a Vitali set is a subset $V$ of $[0,1]$ such that the restriction to $V$ of ...
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Locally compact Polish groups acting on standard Lebesgue spaces

If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...
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Does every cofinal branch through Kleene's O compute true arithmetic?

My question concerns cofinal branches through Kleene's $O$, which is a set of natural numbers and a computably enumerable relation $<_O$ on this set that provides ordinal denotations for any ...
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Is there a simple proof that proves $C^1[0, 1]$ is $\Sigma^1_1$ in $C[0, 1]$?

In his book, "Descriptive Set Theory", Moschovakis states $C^1[0, 1]$ is $\boldsymbol{\Sigma}^1_1$ in $C[0, 1]$ in the exercise 1E.8. Here, $C[0, 1]$ is the space (metrized by the sup norm) of ...
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Function whose graph is a Borel relation

Suppose $f\colon\mathbb{R}^{\omega}\longrightarrow\mathbb{R}$ is a function such that $$G(f):=\{(x,y)\in\mathbb{R}^{\omega}\times\mathbb{R}\mid f(x)=y\}$$ is a Borel set. Does it necessarily follow ...
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187 views

Borel hierarchy and tail sets

Let $A$ be a finite set, and let $A^\infty$ be the set of all sequences $(a_n)_{n=1}^\infty$ of elements of $A$. A set $B \subseteq A^\infty$ is a tail set if for every two sequences $\vec a, \vec b \...
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113 views

Lambda system generated by a non-atomic collection

Consider a probability space $(X,\Sigma,P)$. Let say that a collection $\mathcal{B}\subseteq\Sigma$ is non-atomic if for every $E\in\mathcal{B}$ and $\alpha\in(0,P(E))$ there exists $F\in\mathcal{B}$ ...
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Is the co-projection of the projection of a Borel set Lebesgue measurable?

Reading the classical book of Kechris "Classical descriptive set theory", I found the following facts The projection of a Lebesgue measurable set need not be Lebesgue measurable, but the projection ...
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235 views

Pointwise definable models of determinacy

Suppose that the Axiom of Determinacy (AD) holds in $L(ℝ)$, and for some statement φ, $α$ is minimal such that $L_α(ℝ)⊨φ$. Do definable (in $L_α(ℝ)$) elements of $L_α(ℝ)$ form an elementary ...
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Reference for Borel $\sigma$-algebra of topology of convergence in probability

I'm pretty sure I can prove the "Theorem" given further below (without very much difficulty), but it seems way too basic not to have been noticed before. So I'm wondering if there are any papers/...
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When is validity definable in $L_\alpha$?

Below, $\alpha$ is a countable p.r.-closed ordinal $>\omega$. Let $\mathcal{L}_\alpha=\mathcal{L}_{\infty,\omega}\cap L_\alpha$ (note that this is not the same as $\mathcal{L}_{\omega_1,\omega}\...
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The measure of ideals generated by random reals

We assume that for every real $x$, $L[x]$ only contains countably many reals. Given a set $X$ of reals, then $L$-ideal generated by $X$ is the smallest set $I$ of reals so that For any reals $x\in ...

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