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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Strong form of $\mathtt{PSP}$ for $K_\sigma$ sets

Consider an uncountable perfect $K_\sigma$ set $X\subseteq \omega^\omega$, where $K_\sigma$ means countable union of compact sets, perfect means that $X$ has no isolated points and $\omega^\omega$ is ...
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For measure-preserving systems, is countable generatability of the invariant $\sigma$-algebra equivalent to almost all points being periodic?

Let $X$ be a second countable Hausdorff topological space, let $T \colon X \to X$ be a Borel-measurable map, define the $\sigma$-algebra $\mathcal{I}=\{A \in \mathcal{B}(X) : T^{-1}(A)=A\}$, and for ...
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3 votes
1 answer
84 views

$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$

Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set". A priori this sentence is weaker than the ...
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3 votes
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Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
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Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$ Their ...
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3 votes
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62 views

Forcings that preserve $\mathtt{PSP}$

By $\mathtt{PSP}$ I mean the statement "every subset of the reals has the perfect set property, i.e. either is countable or it contains an homeomorphic copy of the Cantor space $2^\omega$". ...
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5 votes
1 answer
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Shelah's "Can you take Solovay's inaccessible away?"

I was wandering if there was a book, thesis or some notes where Shelah's argument for $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...
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3 votes
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150 views

Existence of a non-$Q$-set without the perfect set property

We have the following theorem: Suppose $\omega_1^L=\omega_1$ then there exists a $\Pi_1^1$ subset of reals without the perfect set property Moreover, under the same hypotheses, we can prove actually ...
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3 votes
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108 views

Co-analytic $Q$-sets

A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
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4 votes
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$\mathtt{PSP}$ implies the consistency of inaccessible cardinals

I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}...
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2 votes
1 answer
114 views

Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$

My question is: Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$? Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\...
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4 votes
1 answer
122 views

Comparing bornologies for domination/escaping

Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$: $\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\...
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4 votes
1 answer
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Are there such a complete metric space X of weight k (w(X)=k) and ....?

Are there such a complete metric space $X$ of weight $k<\mathfrak{c}$ ($w(X)=k$) and a family $\{F_{\alpha}: \alpha<k\}$ of closed subsets of $X$ that $k<|X\setminus \bigcup F_{\alpha}|<\...
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3 votes
0 answers
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What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?

A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq ...
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4 votes
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107 views

Separable metrizable spaces far from being completely metrizable

I came across a kind of separable metrizable space that is "far" from being completely metrizable. Before specifying what I mean with "far", I recall that a space is said to be ...
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6 votes
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Failure of Baire's grand theorem when the hypothesis is weakened to separable metric space

The statement of Baire grand theorem gives a characterization of Baire class 1 functions between a completely metrizable separable space (aka Polish space) and a separable metrizable space. The ...
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4 votes
1 answer
128 views

Hedgehog of spininess $κ$ is an absolute retract?

Let $κ$ be an infinite cardinal, $S$ a set of cardinality $κ$, and let $I = [0, 1]$ be the closed unit interval. Define an equivalence relation $E$ on $I × S$ by $(x,α) E (y,β)$ if either $x = 0 = y$ ...
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4 votes
0 answers
173 views

Is there an uncountable family of "hereditarily unembeddable" continua?

Define a family $\{C_i\}_{i\in I}$ of continua, that is compact connected metrizable spaces, to be hereditarily unembeddable (a name I just made up) iff for all $i\neq j$ no nontrivial subcontinuum of ...
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1 vote
0 answers
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Separately continuous functions of the first Baire class

Problem. Let $X,Y$ be (completely regular) topological spaces such that every separately continuous functions $f:X^2\to\mathbb R$ and $g:Y^2\to \mathbb R$ are of the first Baire class. Is every ...
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1 vote
1 answer
60 views

Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$

A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' Classical Descriptive Set Theory, p. 192): Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\...
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1 vote
2 answers
209 views

Subsets of the Cantor set

A copy of the Cantor set is a space homeomorphic to $2^{\omega}$. Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\...
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12 votes
0 answers
193 views

Pointwise convergence of trigonometric series

$f$ is said to have trigonometric expansion if some series $\sum_{n\in\mathbb{Z}}c_ne^{inx}$ converges pointwise to $f(x)$. On the second page of the article Trigonometric series and set theory, ...
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3 votes
0 answers
162 views

Set-theoretic hierarchy using the uniqueness quantification

Has an equivalent of the set-theoretic hierarchies (arithmetical, hyperarithmetical, Levy etc.) that uses the uniqueness quantification, $\exists !$ (and its dual, $\neg\exists!\neg$) been studied ...
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2 votes
0 answers
117 views

A question about Shoenfield's absoluteness theorem and $0^{\sharp}$

Shoenfield's theorem states that any $\Pi_{3}^{1}$ sentence that holds in $V$ holds in $L$, but I know that it is consistent with ZFC that there exists a set $A\subset \mathbb{N}$ where $A\not\in L$ ...
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5 votes
1 answer
177 views

On the chromatic number of an analytic graph

Let $X$ be a Polish space and let $G\in\mathbf{\Sigma}^1_1(X^2)$ be a graph on $X$, that is an irreflexive and symmetric relation on $X$. Given a cardinal $\kappa$ we say that $G$ has chromatic number ...
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4 votes
1 answer
155 views

Consistency of the Hurewicz dichotomy property

Just to fix the environment, let's work in the Baire space $\omega^\omega$, the space of infinite sequences of natural numbers with the product of the discrete topology over $\omega$. We say that a ...
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4 votes
1 answer
169 views

Borel ranks of Turing cones

For a non recursive $x \in 2^{\omega}$, define $C_x = \{y \in 2^{\omega}: x \leq_T y\}$. Note that $y \in C_x$ iff $(\exists e)(\forall n)(\Phi^y_e(n) = x(n))$ where $\Phi_e$ is the $e$th Turing ...
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5 votes
0 answers
266 views

What is known about when regularity properties only hold for partial boldface pointclasses?

Apologies in advance for a rather vague and open-ended question. Results about regularity properties of the projective pointclasses tend to have a wholesale flavor. By this I mean one tends to be ...
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2 votes
0 answers
176 views

A question about infinite product of Baire and meager spaces

Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space. Does anyone have any suggestions to demonstrate Proposition 1? I was ...
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3 votes
1 answer
219 views

A submodel of set theory with all reals which every set is analytic

This is a continuation of my previous question. Recall that a subset $A \subseteq {}^\omega\omega$ is analytic if it is the continuous image of the Baire space. I would like to know if there exist two ...
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3 votes
2 answers
373 views

A Baire subset of reals that is not Suslin measurable

EDIT: The definition of a Suslin measurable set I wrote here is incorrect. It should be that $\mathcal{S}$ contains the field (or algebra) of open subsets of ${}^\omega\omega$ (or, in other words, it ...
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7 votes
2 answers
195 views

Hausdorff quasi-Polish spaces

A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. ...
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2 votes
0 answers
78 views

Codimension of analytic linear subspaces in Polish vector spaces

Let $A$ be a linear analytic subspace of a Polish vector space $X$. Using Piccard-Pettis Theorem, it is easy to prove that $A$ has uncountable codimension in $X$. If $A$ is of type $F_\sigma$, then ...
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3 votes
0 answers
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Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
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10 votes
0 answers
215 views

What is the smallest $\sigma$-algebra of reals that is closed under addition of sets?

What is the smallest $\sigma$-algebra $\Sigma\subseteq\mathcal P(\Bbb R)$ containing the open sets and such that if $A,B\in\Sigma$, then $$A+B=\{a+b\mid a\in A,b\in B\}\in\Sigma?$$ I know that neither ...
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2 votes
1 answer
171 views

How much choice is needed to prove this statement?

Consider the following statement (in $\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$): There exists $(C_\alpha, x_\alpha)_{\alpha \in \omega_1}$ s.t. $C_\alpha \subseteq \mathbb{N}^\mathbb{N}$ is closed ...
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9 votes
1 answer
512 views

How much choice is necessary to prove this statement?

Consider the following statement (in $\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$): There exists $(\varphi_\alpha)_{\alpha\in\omega_1}$ with $\varphi_\alpha : \alpha \rightarrow \mathbb{N}^\mathbb{N}$ ...
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1 vote
0 answers
76 views

Separating two sets by a $\boldsymbol{\Delta}_3^0$ set

Let $X$ be a Polish space and $A,B\subseteq X$ be such that $A\cap B = \emptyset$, we know that if there is no $\boldsymbol{\Delta}_2^0$ set separating $A$ from $B$ then there exists a Cantor set $C\...
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  • 753
2 votes
1 answer
86 views

Is every Borel function a projection of a Borel function with closed graph?

Is it true the following statement? Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$, there exists a Polish space $Z$ and a Borel function $g:X \rightarrow Y\times Z$ with closed ...
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7 votes
1 answer
280 views

In a Polish space, is every analytic set the continuous image of a Borel set from the same Polish space?

I'm confused by a subtle point in the definition of analytic sets. Suppose I have a Polish space $X$. Now I start with the collection of Borel sets in $X$ and take all their continuous images in $X$. ...
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6 votes
1 answer
244 views

A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16): Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
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2 votes
0 answers
124 views

Are there any measurable spaces of functions

I am approaching this question from a probability perspective, and am hoping for some kind of framework to help understand all of this. I believe I may have even asked a similar question on here in ...
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  • 221
6 votes
1 answer
201 views

Existence of a winning strategy in the $\boldsymbol{\Delta}_2^0$ game

Consider the following infinite perfect information game with two players (the name I gave in the title of the post it totally made up): at each round $i \in \omega$, player $\mathrm{I}$ picks a ...
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  • 753
3 votes
0 answers
231 views

Borel equivalence relations on Ellentuck cubes

Is there a Borel equivalence relation $E$ on $[\omega]^\omega$ such that $E \not \leq_B E_0$ and for any $a \in [\omega]^\omega$ we have that $E \upharpoonright [a]^\omega$ is Borel bireducible with $...
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4 votes
0 answers
106 views

Is every analytic set the projection of a set with sections large in some sense?

Is every analytic set $A$ in, say, $I=[0,1]$, the projection of a Borel set $B$ in, say, $[0,1] \times [0,1]$, $A = \pi_1(B)$, with the following property: For every regular Borel probability measure $...
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7 votes
0 answers
270 views

Which countable ordinals are "Barwise compact" for $\mathcal{L}_{\infty,\omega_1}$?

Barwise compactness says (as a special case) that whenever $\alpha$ is countable and admissible, $T\subseteq\mathcal{L}_{\infty,\omega}\cap L_\alpha$ is $\alpha$-c.e., and every subset of $T$ which is ...
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24 votes
2 answers
909 views

Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$

For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary ...
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7 votes
0 answers
386 views

Infinite time Turing machines, semi-decidable sets and descriptive set theory

Definition A set of reals $A$ is said to be ittm-eventually-semi-decidable if there is an Infinite Time Turing Machine programme $P_e$ so that $x\in A$ iff $P_e(x)$ has converged on “1” on its ...
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  • 4,701
7 votes
1 answer
278 views

Uniformization under AD

Can the following uniformization statement be proved by $ZF+AD+DC$? For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable ...
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  • 3,771
5 votes
0 answers
166 views

Can maximal filters of nowhere meager subsets of Cantor space be countably complete?

Let $X$ denote Cantor space. A subset $A\subseteq X$ is nowhere meager if for every non-empty open $U\subseteq X$, we have $A\cap U$ non-meager. We call $\mathcal{F}\subseteq \mathcal{P}(X)$ a maximal ...
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