# 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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### $\mathtt{PSP}$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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 ...