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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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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Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]

Introduction Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. I want to define an average on $f$ using a new measure and integral related to the Lebesgue Measure. The Lebesgue ...
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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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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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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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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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Borel rank collapse in Hilbert cube modulo $\sigma$-ideal generated by zero-dimensional sets

Both of the commonly studied $\sigma$-ideals (meager sets and null sets) in Polish spaces with a natural measure (i.e. $\mathbb{R}$, $[0,1]$, $[0,1]^\omega$, $2^{\omega}$, etc.) have the nice property ...
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Detecting comprehension topologically

This question basically follows this earlier question of mine but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA_0$. For $X$ a Turing ideal we get the map $c_X$...
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Definition of random measures

Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from? I would like to have a motivation ...
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Kurtz randomness and supermartingales with infinite *limit*

Suppose you replace the usual success conditions for a supermartingale (lim sup is infinite) with the requirement that the actual limit is infinite, e.g. a supermartingale $B$ succeeds on $X \in 2^\...
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Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?

Every $Π^1_1$ formula $φ$ without free second order variables can be converted into a $Σ^1_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, ...
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General theory of the reals in Solovay-like models

Solovay's model is a famous model of $\sf ZF$ where we start in $L$ with $\kappa$ inaccessible, and we collapse all the ordinals below $\kappa$ to be countable, without collapsing $\kappa$ itself, and ...
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Undetermined Banach-Mazur games in ZF?

This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question. Given a ...
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Suslin representation of sets and limits to Shoenfield's Absoluteness

For any natural number $k \in \omega$ and any set $X$ the set $$T \subseteq \bigcup_{m \in \omega} (\omega^m)^k \times X^m $$ is a tree on $\omega^k \times X$ iff $$(t_o, \ldots, t_k) \in T \: \...
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Improving a Lindstrom-y fact about $\mathcal{L}_{\omega_1,\omega}$?

See e.g. the last section of Ebbinghaus/Flum/Thomas for the relevant background on abstract model theory. Below, all languages are finite for simplicity. "$HC$" is the set of hereditarily countable ...
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270 views

“Robinson arithmetic” for (some) levels of $L$?

I'll write "$\mathcal{L}_\alpha$" for the fragment $\mathcal{L}_{\infty,\omega}\cap L_\alpha$. Say that a countable admissible $\alpha$ is Robinsonian if there is some sentence $\varphi\in\mathcal{L}...
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When does “sufficient genericity” actually suffice?

Fix a forcing notion $\mathbb{P}$. Say that a formula $\varphi(x)$ with parameters is $\mathbb{P}$-enforceable if there is some countable set $\mathcal{D}$ of dense sets in $\mathbb{P}$ such that for ...
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Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional

A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$. A ...
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Question about additive subgroups of the real line and the density topology

I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question. Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $...
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Can it be that universal measurability is preserved by projections?

I am currently discovering descriptive set theory—with much pleasure! It is something of a surprise to me that, while the Borel hierarchy is indexed by $\omega_1$, the projective hierarchy is only ...
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How much choice is required for a countably-infinite index subgroup of the real additive group?

The existence of such subgroups implies the existence of a non-measurable set; simply intersect each of the cosets with $[0,1]$. The results will all have equal outer measure, but their union will be ...
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Source on smooth equivalence relations under continuous reducibility?

This question was asked and bountied at MSE, but received no answer. In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
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Is the inverse of a measurably parametrised family of bijections between standard Borel spaces measurably parametrised?

It is known that a measurable bijection $f \colon [0,1] \to [0,1]$ has a measurable inverse. (Here, all measurability is simply with respect to the Borel $\sigma$-algebra of $[0,1]$.) Now fix an ...
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Ramsey type properties of $F_\sigma$ ideals

Let $I \subseteq 2^\omega$ be any $F_\sigma$ ideal containing every finite sets : $\forall X \in I\ \forall Y \subseteq X\ \text{ we have } Y \in I$ $\forall k\ \forall X_1,\dots,X_k \in I\ X_1 \cup \...
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The Borel class of a countable union of $G_\delta$-sets, which are absolute $F_{\sigma\delta}$

Problem. Assume that a metrizable separable space $X$ is the countable union $X=\bigcup_{n\in\omega}X_n$ of pairwise disjoint $G_\delta$-sets $X_n$ in $X$ such that each $X_n$ is an absolute $F_{\...
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Borel / Wadge hierarchies on subsets closed under prepending a finite prefix

I'm interested in subsets $X$ of the Cantor space ($2^\omega$) or the Baire space ($\omega^\omega$) that are closed under prepending an arbitrary finite prefix: $$ (x_1, x_2, \dots) \in X \implies (...
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Uncountable disjoint closed coverings of $[0,1]$

It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this math.stackexchange question. The proof ...
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Set of perfect subsets of a Borel set

Let $\mathbb{P}$ be the set of all perfect (i.e., every node has incomparable successors) subtrees of the full binary tree $2^{<\omega}$. We can endow $\mathbb{P}$ with a Borel structure by ...
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Fraïssé Limit and Ultraproduct

Let $\mathcal{C}=\{M_i\}$ be a Fraïssé class of finite $\mathcal{L}$-structures with the generic model $M$. Also, let $M^*=\prod_U M_i$ the be ultraproduct of members of $\mathcal{C}$ where $U$ is a ...
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Regularity properties of Turing-invariant and arbitrary sets of reals

The question whether Turing determinacy implies $AD$ is a well-known open problem. I was wondering if anything is known about the following analogous question: Let $\Gamma$ be a regularity property (...
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Convergence and winning strategies

Suppose we have a set $B\subseteq 2^\omega\times\omega^\omega$ and a sequence $(x_n)$ in $2^\omega$ such that for each $n$, Player I (the one trying to get into the payoff set) has a winning strategy ...
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A slight extension of Sacks theorem

Sacks proves the following theorem first. Theorem 1: If $\alpha$ is a countable admissible ordinal, then there is a real $x$ so that $\omega_1^x=\alpha$. Anyone knows who proves the following ...
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A partial relativization of Gandy's basis theorem

Gandy's basis theorem says that any nonempty $\Sigma^1_1$ set $A$ contains a real $x$ with $\omega_1^x=\omega_1^{CK}$, the least nonrecursive ordinal. Now the following question seems quite ...
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Variously pointed closed sets

A tree $A\subseteq \omega^{<\omega}$ - possibly with dead ends - is pointed iff every path $p\in[A]$ has $p\ge_TA$. This lifts to two distinct notions of pointedness for closed sets in Baire space: ...
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Do the higher levels of the Borel hierarchy correspond to absolute topological properties?

It is well known that a subset $Y$ of a Polish space $X$ is completely metrisable iff it is a $G_\delta$ subset. This relates a relative topological property of the subspace $Y \subset X$ to an ...
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Abstract transverse measure theory

After reading Noncommutative Geometry book (see here) I came across the notion of the so called abstract transverse measure theory which is a generalization of standard measure theory well adapted to ...
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1answer
180 views

Examples of independent $\Sigma_4^1$ statements

As in the title, I'm looking for examples of $\Sigma^1_4$ (preferably complete) sentences which are independent from ZFC in both ways, namely given a model $V$ we can extend it to $V'$ where such a ...
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Nowhere Baire spaces

Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space $X$ nowhere Baire ...
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1answer
147 views

Product of Bernstein sets

Remember that a Bernstein set is a set $B\subseteq \mathbb{R}$ with the property that for any uncountable closed set, $S$, in the real line both $B\cap S$ and $(\mathbb{R}\setminus B)\cap S$ are non-...
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Concerning Luzin-(N)-property

Definition: a function $f:\mathbb{R}\to \mathbb{R}$ has Luzin-(N)-Property if $f$ maps any null set to a null set. By https://www.encyclopediaofmath.org/index.php/Luzin-N-property, it is known that ...
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310 views

Is there a universally meager air space?

Let $\mathcal P$ be a family of nonempty subsets of a topological space $X$. A subset $D\subset X$ is called $\mathcal P$-generic if for any $P\in\mathcal P$ the intersection $P\cap D$ is not empty. ...
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Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...

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