# 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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### 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 ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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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### 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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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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### 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 ...
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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### 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 ... 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 $\alpha$. ... 45 views

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 $\{F \in F(X) : F \cap U \neq \... 0 votes 0 answers 37 views ### Homeomorphic product spaces In his text book on descriptive set theory, Proposition 25.2, Kechris implicitely uses the fact that$(\omega \times \omega)^\omega$and$\omega^\omega \times \omega^\omega$are homeomorphic, where$\...
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 ...
### 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 ...