Questions tagged [descriptive-set-theory]

In descriptive set theory we mostly study Polish spaces such as the Baire space, the Cantor space, and the reals. Questions about the Borel hierarchy, the projective hierarchy, Polish spaces, infinite games and determinacy related topics, all fit into this category very well.

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Under what minimum assumptions a null set is necessarily meager?

$A\subset \Bbb{R}$ is a null set if $\lambda(A) =0$ ($\lambda$ :Lebesgue measure) $A\subset \Bbb{R}$ meager if $A$ is countable union of nowhere dense sets (sets whose closure contains no nonempty ...
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Can we find $f\in \Bbb{R}^{[0, 1]}$ with the property $\mathcal{M}$ which doesn't satisfy the property $\mathcal{B}$?

$f:[0, 1]\to \Bbb{R}$ be a function. $f$ satisfy the property $\mathcal{M}$ of $f(A) $ is meagre for every $A\subset [0, 1]$ meagre. $f$ satisfy the property $\mathcal{B}$ if $f(A) $ is a set with ...
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Baire class one and Borel functions

The picture below is page 190 of Classical Descriptive Set by Kechris, there is Theorem 24.3, I just need that theorem for Baire class 1 case, in other words, I just need to proof for the case when $\...
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find representative for equivalence relation Discrete math

T = {<X, Y > ∈ P(R) × P(R) | 0 is not ∈ X Symmetric difference Y } someone can explain why this set {{},{0}} is representative of T. I thougt that the stmmetric difference between {},{0} is 0 .....
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Are the open sets of $\{0, 1\}^I$ measurable?

Let $I$ be an uncountable set and let $2^I=\{0, 1\}^I$ be the set of all functions from $I$ into $\{0, 1\}$. Consider the product measure $\mu$ on $2^I$. The domain of this measure is the $\sigma$-...
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Can the quotient space obtained by partitioning the closed interval into Cantor sets be Hausdorff?

In response to this question Can the Interval be Covered by Disjoint Cantor Sets? it was pointed out that the answer is, Yes: see Theorem 1.14 of Paul Bankston and Richard J. McGovern, Topological ...
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Properties about the comeagre null sets in $\mathbb{R}^n$ that is non-Borel

I have learnt from this question that comeagre null sets (sets with Lebesgue measure zero whose complement is a countable union of nowhere dense sets) do exist, and they cannot be included in $F_\...
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Borel Determinacy Theorem in multidimensional case

Let $A$ be a nonempty set and $X \subseteq A^{\mathbb{N}}$. In the classic version game runs in such a way: there are two players (I and II), at the beginning I writes any $a_1 \in A$, then II writes ...
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proof that "reasonably defined" sets in $2^\mathbb{Z}/{\sim}$ have measure $1$ or $0$

I am basing this question on the slides: http://math.yorku.ca/~moliver/how.pdf One is given the set of doubly infinite strings of $0$ and $1$ whose origin is forgotten which is the set of equivalence ...
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Measure and category of codes of countable ordinals

A well-known result from descriptive set theory says that for any countable ordinal $\alpha$, the set of reals that code a well-ordering on $\omega$ with ordertype $\alpha$ is a Borel set. I wonder if ...
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Subsets of Well-ordered Sets

Suppose I) X = $\mathbb{N}$ II) $X_{\alpha}$ = {x $\ge$ $\alpha$ | x $\in$ $\mathbb{N}$} Hence, $\cap_{\alpha}^N$ $X_{\alpha}$ = [N, $\infty$).By II, all $X_{\alpha}$ are well ordered sets with a ...
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Is there a topology $\tau$ on $\mathbb{N}$ such that $(\mathbb{N},\tau)$ is homeomorphic to $(\mathbb{Q}\cap[0,1], \tau_{usual})$?

I would like to show that there exists a topology $\tau$ on $\mathbb{N}$ such that $(\mathbb{N},\tau)$ is homeomorphic to $(\mathbb{Q}\cap[0,1], \tau_{usual})$. Sierpinski’s theorem: Any countable ...
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What does a language parameterizing another language mean?

I am reviewing my class notes, and I came across this expression - The $n$-th slice of $A \subseteq \Sigma^*$ is $A_n = \{x \in \Sigma^* \mid {\langle n,x \rangle} \in A \}$ $C$ parameterizes $D$ (...
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Question about a claim of Bukovský in his 1971 paper "Ensembles génériques d'entiers"

In his paper Ensembles génériques d'entiers, Bukovský says the following (recall that if $M$ is a transitive model of ZF containing all ordinals and $x$ is a subset of $M$, then $M[X]$ denotes the ...
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Model of ZFC Where Every Uncountable Co-Analytic Set of Reals Has a Perfect Subset

I’ve been doing some reading on classical descriptive set theory for fun, and I’m trying to find a reference for the following claim: Theorem: There is a model of ZFC Set Theory in which for every ...
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Tietze extension theorem for $X=Y=\mathbb{R}^\omega$

The Tietze extension theorem says that if $X$ is a Polish space (even a normal space) and $Y=\mathbb{R}^n$, then a continuous function $f:C \rightarrow Y$ on a closed set $C \subseteq X$ can be ...
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1 answer
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Are Borel measurable functions closed in pointwise topology?

Let $X$ be a metrizable space. The Lebesgue-Hausdorff theorem states that the minimal class $\mathcal{C}$ of functions $f :X \to \mathbb{R}$ closed under pointwise limits of sequences, such that $C(X) ...
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Countable topological basis with some inclusion properties

Given a second countable topological space $(X,\tau)$, I want a countable basis $\mathcal{B}$ with the following properties: \begin{align}&\forall B \in \mathcal{B} \text{ the set } \{B' \in \...
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Name for FINITE unions and intersections of closed and open sets? (Reference/terminology)

Given a topological space $(X , \mathcal{O})$, along with the "topology"/"family of open sets" $\mathcal{O}$ one gets (implicitly) the family of closed sets $\mathcal{C}$. Then $\...
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Baire space which doesn't contain a dense completely metrizable subspace

Let us work with separable metrizable spaces. It's known that if $X$ contains a dense completely metrizable subspace then $X$ is Baire. Moreover, if we assume that $X$ embeds into any space as a Borel ...
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A possible easy proof of Borel determinacy?

I was initially trying to prove determinacy of $\mathbf{\Sigma}^0_2$ games, but surprisingly the proof I came up with seems to generalise all the way to Borel determinacy. Obviously this sounds a ...
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Verify $F_\sigma$ of a countable sum of functions

For each $n\in\Bbb Z$, let $f_n\colon [n,n+1]\to\Bbb R$ be a function such that $f_{n}(n+1)=f_{n+1}(n+1)$ and $f_n$ is $0$ on $\Bbb R\setminus [n,n+1]$ and $f^{-1}_{n}(U)$ is $F_\sigma$ ( a countable ...
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Countable sum of Baire of class 2 functions on $\Bbb R$

For every $k\in\Bbb Z$, let $f_k$ be a Baire 2 class function on $\Bbb R.$ Assume $\sum_{k\in\Bbb Z} f_k$ is convergent. Define $f:=\sum_{k\in\Bbb Z} f_k$ so $f$ is a function. Moreover, $f$ is a ...
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Can we classify all topological space $(X, \tau) $ where every second category sets are Residual sets?

$(X, \tau) $ be a topological space. $A\subset X$ is Residual if $X\setminus A$ is of first category. In a Baire space, a Residual set is of second category. $A\subset X$ Residual, then $X\setminus A$ ...
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Cardinality bound of a sigma algebra without transfinite induction or (full) axiom of choice.

The Question Let $A$ be a set. It is true that there exists some cardinality $\kappa_A$ such that whenever $\Sigma$ is a $\sigma$-algebra, and $\Sigma$ contains a generating set of cardinality $|A|$, ...
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6 votes
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Measurable invariance of domain

Invariance of domain theorem tells us that if a subset $V$ of $\mathbb{R}^n$ is homeomorphic to an open subset of $\mathbb{R}^n$, then $V$ must be open itself. Question: If a subset $V$ of $\mathbb{R}...
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How to prove that the following metric induces the subspace topology?

I am trying to follow Theorem (3.11) of Kechris's Classical Descriptive Set Theory. In this part of the proof he shows that a $G_{\delta}$-subspace Y of a completely metrizable space $(X,d)$ is ...
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4 votes
1 answer
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Is there a function whose graph is connected but contains no arcs?

Is there $f:\mathbb{R}\to\mathbb{R}$ such that the graph is connected but for any two points in the graph, there isn´t any path in the graph between them? I´m not sure what the answer is going to be. ...
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2 votes
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Small index property

The following is an excerpt of Truss's 1989 paper Infinite Permutation Groups II.: Subgroups of Small Index. (The transcript will appear later.) Here, $2^\omega$ the Cantor space, automorphisms mean ...
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The set of all points belonging to infinitely many closed sets not an $F_{\sigma}$

Let $\{E_n\}_{n\ge 1}$ be a countably infinite family of closed subsets of R. Let $A$ be the set of points which belong to infinitely many of the sets $E_n$. Then it is known that $A=\bigcap_{n=1}^{\...
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7 votes
1 answer
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Kunen exercise IV.4.13 (4): Topological version of effective AC

I am dealing with Kunen's The Foundations of mathematics exercise IV.4.13 (4): Let $X$ denote the Cantor set. Prove if $S\subset X\times X$ is open, then there is an $F\subset S$ such that $F$ is the ...
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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. Pure ...
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5 votes
4 answers
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Why not we avoid the phrase "if we assume AC " and take it as granted?

This is slightly different question. First I need to mention that I am neither a mathematician nor a researcher. As an ordinary student the separation " with and without Axiom of choice " ...
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2 votes
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Is it true that, for any $X$ Polish uncountable, every boldface class of $X$ is lightface with respect to some oracle?

I am wondering the question in the title: let $X$ be uncountable Polish. Consider the standard Borel structure on $X$; that is, $\mathbf{\Sigma}_1^0(X)$ are the open sets, etc.. Is it true that, with ...
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Products of Product Spaces

Let $\mathbb{N}$ be Natural numbers set with the discrete topology $\tau_{D}$ induced by the discrete metric. Then $\mathbb{N}$ copies of $(\mathbb{N},\tau_{D})$ gives us the Baire space $(\mathbb{N}^{...
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4 votes
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Is there any condition that makes a measure zero set countable?

Countable set: A set $X$ is said to be countable if there exists a one-to-one mapping $f:X\to \mathbb{N}$. Measure Zero set : A set $X$ with zero Lebesgue measure. First Category : A metric space $(...
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1 vote
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A continuous function with uncountable image between Polish spaces is injective on a generic compact set

Exercise 8.8ii in Kechris Classical Descriptive Set Theory asks to prove that if $f\colon X\to Y$ is a continuous function between Polish spaces such that $f(X)$ is uncountable, then there is a Cantor ...
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Recursive in a $\Sigma^0_2$-singleton implies recursive

I've encountered two related remarks that I can't figure out. They are: If a real number is recursive in a $\Sigma^0_2$-singleton, then it is recursive; This is best possible, as $\{\emptyset^{(\...
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1 vote
1 answer
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Gentle Introduction to Descriptive Set theory

If find the common literature on Descriptive Set theory pretty tough. Is there a gentle recommandation? I would really like to learn this topic on my own, so I am searching for something diguestible ...
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1 answer
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Why is $A^\mathbb{N}$ with the discrete topology a polish space

I am currently preparing for a part of a seminar in topology/descriptive set theory and am working with the A. Kechris' book. I am confused about some results for one of the easier examples, the ...
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2 votes
2 answers
161 views

applying Martin's axiom

I am struggling with the following problem: Let $A_n$ for $n \in \omega$ be sets of respective cardinality $n + 1$, and let $X = \prod A_n$. Show that Martin's axiom for $\kappa$ proves that for every ...
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2 votes
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Separating preimages of opens sets by Baire functions

Given Baire class $1$ function $f:\omega^\omega\rightarrow \omega^\omega$ I want to prove that for every non empty $U_1,U_2$ open sets with $U_1\cap U_2 = \emptyset$, there exists a $\boldsymbol{\...
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Where do winning strategies occur for Player II in the Determinacy of Computable Open Games relative to a parameter?

Moschovakis goes over various theorems proving the Determinacy of closed/open games, and I am reading into various papers regarding the characterization of ordinals where winning strategies for the ...
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1 vote
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Sequences $(A_n)_{n\in\omega}\subset \boldsymbol{\Delta}_2^0$ such that $\bigcup_{n\in\omega} A_n \in \boldsymbol{\Delta}_2^0$

Given a Polish space $(X,\tau)$ can we characterize in some meaningful way the sequences $(A_n)_{n\in\omega}$ of $\boldsymbol{\Delta}_2^0(X)$ sets (we can assume them to be pairwise disjoint) such ...
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1 vote
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numerical formulas over $V$ and $L$

Let $\phi$ be a numerical formula, i.e., all variables range over naturals. Then we know that $\phi^{V} = \phi^{L}$, where $V$ is the universe and $L$ the constructible sets. Our previous equality ...
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1 vote
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Ordinals in $L$, the constructible universe

I am trying to understand the constructible universe $L$. Based on the way it is constructed, it is clear that every ordinal is included in $L$, i.e., $\alpha \subset L$ for any ordinal $\alpha$. ...
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2 votes
1 answer
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Preimage measurable w.r.t. Effros Borel structure

For a Polish space $X$ let $F(X)$ the set of all closed subsets of $X$. The space $F(X)$ can be equipped with the $\sigma$-algebra generated by the sets of type $\{F \in F(X) : F \cap U \neq \...
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Homeomorphic product spaces

In his text book on descriptive set theory, Proposition 25.2, Kechris implicitely uses the fact that $(\omega \times \omega)^\omega$ and $\omega^\omega \times \omega^\omega$ are homeomorphic, where $\...
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5 votes
1 answer
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What is the Borel Hierarchy?

I'm taking a measure theory class, and our professor mentioned in passing that the Borel sets "stratify" into a hierarchy. But I'm getting lost in the $\Pi$s, $\Sigma$s, and $\Delta$s on the ...
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2 votes
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Overall confusion in Moschovakis' Proof that $\Sigma_2 ^0$ games are determined (Page 221)

I'm reading through Moschovaki's proof that all $\Sigma_2^0$ games are determined, and the second part of the proof is confusing me. I follow up to the point where they prove $u\in W^{\xi}\implies $ I ...
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