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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The cardinality of projections of subsets of the Hilbert cube by inner products

I have three related questions. Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
Boaz Tsaban's user avatar
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5 votes
1 answer
114 views

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?

Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
Alexander Osipov's user avatar
2 votes
1 answer
164 views

Is there a metric separable space with the following properties...?

Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$. Is there a metric separable space $X$ with the following properties: $|X|\geq\...
Alexander Osipov's user avatar
6 votes
0 answers
132 views

Complexity of transfinite 5-in-a-row and other games

Suppose that 5-in-a-row is played on an infinite board, and after an infinite number of moves, if no one won yet and there is an empty square, the game just continues. At limit steps, it is the first ...
Dmytro Taranovsky's user avatar
6 votes
1 answer
201 views

Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$

Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
David Fernandez-Breton's user avatar
4 votes
0 answers
102 views

Closure of a pointclass under universal real quantification

Let us assume $\mathsf{AD}^+$ and let $\Gamma$ be a pointclass such that $P(\mathbb{R})\cap L(\Gamma)=\Gamma$ and $L(\Gamma)\models\mathsf{AD}_\mathbb{R}+\mathsf{DC}$. Since the cofinality of $o(\...
Obrad Kasum's user avatar
4 votes
0 answers
132 views

Proof of: No rapid filter is Lebesgue measurable

I'm studying the following theorem in (Schindler, 2014: Set Theory Exploring Independence and Truth), p. 178-180: Theorem 9.16 (Mokobodzki) No rapid filter F $\subset$ ${}^\omega 2$ is Lebesgue ...
Caro Meier's user avatar
4 votes
1 answer
198 views

Is every compact, sober, second-countable space the image of $2^\omega$?

As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$? As a further bonus, can we strengthen "image" to "quotient"? My motivation for ...
Robin Saunders's user avatar
4 votes
1 answer
236 views

Does every (Abelian) Polish group have a nontrivial locally compact subgroup?

The question is pretty much in the title, suppose that $G$ is an (Abelian) nontrivial Polish group, must $G$ have a nontrivial locally compact (in the induced topology, hence necessarily closed) ...
Alessandro Codenotti's user avatar
8 votes
0 answers
202 views

The Hausdorff dimension of the set of reals of inner models

Suppose that both $M$ and $N$ are models of $ZFC$ with $M\subseteq N$ so that $M$ is definable in $N$. Question Can $(\mathbb{R})^M$ have Hausdorff dimension strictly between $0$ and $1$ in $N$? How ...
喻 良's user avatar
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2 answers
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Follow up question: Shelah's "Can you take Solovay's inaccessible away?"

In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated: Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...
C_M's user avatar
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2 votes
2 answers
238 views

Erdős–Sierpiński duality in locally compact Polish groups (e.g. $\mathbb{R}^n$)

Erdős–Sierpiński mapping for a locally compact Polish group $G$ is a bijection $f$ from $G$ to $G$ such that $A$ is a null set in $G$ with respect to the Haar measure if and only if $f(A)$ is a meager ...
Nugi's user avatar
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6 votes
1 answer
277 views

A variation on pinned equivalence relations

Recall (see e.g. Zapletal, Pinned equivalence relations) that a Borel equivalence relation $E$ on $\omega^\omega$ is pinned iff for every forcing $\mathbb{P}$ and every $\mathbb{P}$-name $\nu$ we have ...
Noah Schweber's user avatar
7 votes
3 answers
425 views

How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?

So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$. Basically, dependent choice on $\mathbb{R}$ says ...
Alex Appel's user avatar
5 votes
1 answer
150 views

Existence of Borel uniformization for coanalytic set with non-$K_\sigma$ sections

Suppose that $X$ is a Polish (or standard Borel) space and $\omega^\omega$ is the Baire space of all natural number sequences. My question is: If $A\subseteq X\times \omega^\omega$ is a coanalytic set ...
Iian Smythe's user avatar
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27 votes
1 answer
2k views

Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
Pietro Majer's user avatar
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1 vote
0 answers
81 views

Approximating evalutation maps at open sets over invariant measures

Let $G$ be a group acting by homeomorphisms on a compact metrizable space, say $X$; let's denote by $\alpha:G\to\mathrm{Homeo}(X)$ the action, $g\mapsto\alpha_g$, and consider the weak-$^*$ compact ...
J G's user avatar
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3 votes
0 answers
163 views

Can the set of parafinite congruences be descriptive-set-theoretically complicated?

Fix an algebra $\mathfrak{A}$ with underlying set $\mathbb{N}$ and finite language $\Sigma$. The set of congruences on $\mathfrak{A}$ is a closed subset $C_\mathfrak{A}$ of $2^\mathbb{N}$ (with the ...
Noah Schweber's user avatar
25 votes
2 answers
2k views

Writing a function on $\mathbb{R}$ as a sum of two injections

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
Burak's user avatar
  • 4,105
4 votes
3 answers
363 views

Hyperarithmetically least elements in $\Pi^1_1$ sets

My question is: Do we have a hyperarithmetically $\le_H$-least real in any $\Pi^1_1$ set? That is Question. Suppose that $A$ is a non-empty $\Pi^1_1$ set. Then can we find a real $a\in A$ such that $...
Hanul Jeon's user avatar
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10 votes
1 answer
374 views

Two dimensional perfect sets

Consider the following family of sets $$ \begin{align*} \mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
Lorenzo's user avatar
  • 2,134
-1 votes
1 answer
121 views

What is an "open Baire set"?

In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
i like math's user avatar
8 votes
1 answer
335 views

"Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton? In more ...
Noah Schweber's user avatar
2 votes
1 answer
106 views

Can convergence in distribution necessarily be realised by almost-sure convergence?

Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each ...
Julian Newman's user avatar
2 votes
0 answers
200 views

The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals

With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...
Ember Edison's user avatar
2 votes
1 answer
243 views

Inner model for KP and a Well-Ordering of the Reals

It is well known that Gödel proved the following theorem: $\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison) So: Is there an inner model for KP/Z/....
Ember Edison's user avatar
10 votes
1 answer
219 views

How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?

This is in some sense a follow-up to this question. The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the ...
James Hanson's user avatar
  • 10.3k
1 vote
1 answer
61 views

Borel sets in Vietoris topology

Let $\mathcal{K} = \mathcal{K}(\mathbb{N}^{\mathbb{N}})$ be the set of all non-empty compact subsets of the Baire space $\mathbb{N}^\mathbb{N}$ equipped with the Vietoris topology. Let $G$ be a Borel ...
Arkadi Predtetchinski's user avatar
7 votes
1 answer
293 views

Strength of Borel determinacy

In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased). Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $...
new account's user avatar
14 votes
0 answers
415 views

Which functions have all the common $\forall\exists$-properties of continuous functions?

This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well. For a ...
Noah Schweber's user avatar
4 votes
0 answers
113 views

Infinite dimensional homology theory for submanifolds of Hilbert and Banach spaces

Is there a version of homology theory for spaces for which explicitly infinite dimensional "cells" are allowed? The spaces in question include e.g. \begin{equation} X = (x: x \in l_2: p_i(x) ...
0x11111's user avatar
  • 473
2 votes
1 answer
163 views

Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product

I've been trying to understand various questions to do with sigma algebras on uncountable product spaces. Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
SBK's user avatar
  • 1,141
3 votes
0 answers
131 views

Which is the smallest $\sigma$-algebra that contains all analytic sets?

Let $X$ be a polish space. Is the smallest $\sigma$-Algebra containing all analytic sets of $X$ (i.e. all subsets $A \subset X$ which are the continuous image of a polish space) the $\sigma$-algebra ...
Joris Wk's user avatar
  • 233
1 vote
0 answers
59 views

Is the projection of an universally measurable set again universally measurable?

Let $(X,\mathcal{A})$ be a measurable space and $(Y,\mathcal{B}(Y))$ be a polish space together with the Borel-$\sigma$-Algebra. There is a Theorem that states: The projection $\pi_X(B)$ of every ...
Joris Wk's user avatar
  • 233
32 votes
2 answers
2k views

Quantifier complexity of the definition of continuity of functions

This was previously asked at MSE, but I was told to ask it on MO. Consider the structure $(\mathbb{R};+,-,*,0,1,<)$. We adjoin to it a unary function $f$ defined everywhere on the set of real ...
user107952's user avatar
  • 2,063
3 votes
1 answer
124 views

A $\Pi^0_1$ class of $\alpha$-subgenerics for all $\alpha < w_1^{CK}$

In Harrington's mimeographed notes (see here) solving McLaughlin's conjecture he builds reals $f \in \omega^\omega$ which have the property of being $\alpha$-subgeneric defined as follows. He does ...
Peter Gerdes's user avatar
  • 2,551
4 votes
1 answer
138 views

Is the set of clopen subsets Borel in the Effros Borel space?

Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
Iian Smythe's user avatar
  • 2,991
3 votes
0 answers
149 views

Image of open set under a real analytic map is Borel?

It is well known that the image of a Borel set under a continuous map may fail to be Borel. This MSE question shows even $C^\infty$ maps may fail to map Borel sets to Borel sets. I want to ask whether ...
Anthony Quas's user avatar
  • 22.4k
2 votes
0 answers
45 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
13 votes
1 answer
445 views

How much determinacy do you need for second order arithmetic to be as strong as ZFC?

From Wikipedia (I couldn't find the original source): $\text{ZFC} + \{\text{there are $n$ Woodin cardinals: $n$ is a natural number}\}$ is conservative over $\text{Z}_2$ with projective determinacy. ...
Christopher King's user avatar
8 votes
1 answer
197 views

Can totally inhomogeneous sets of reals coexist with determinacy?

A special case of a theorem of Brian Scott (from On the existence of totally inhomogeneous spaces) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $...
Noah Schweber's user avatar
6 votes
0 answers
119 views

Detecting uncountable cardinalities, this time with determinacy

By "small cardinality" I mean a Scott cardinality onto which $\mathbb{R}$ surjects ($0$ isn't interesting here). $\mathcal{R}=(\mathbb{R};+,\times,\mathbb{Z})$ is the field of real numbers ...
Noah Schweber's user avatar
4 votes
1 answer
215 views

Comparing bornologies for cardinal characteristics via Borel maps

This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
Noah Schweber's user avatar
1 vote
0 answers
76 views

Existence of a stronger notion of perfect measures

Let $\mathcal{X}$ be a measurable space with its $\sigma$-algebra $\mathcal{B}_\mathcal{X}$ and let $\mathbb{R}$ be the real numbers endowed with its Borel $\sigma$-algebra $\mathcal{B}_\mathbb{R}$. ...
Packo's user avatar
  • 106
6 votes
2 answers
262 views

Atoms for Markov kernels

Let $X$ and $Y$ be standard Borel measurable spaces. A Markov kernel $f : X \rightsquigarrow Y$ is a map $f(-|-) : \Sigma_Y \times X \to [0,1]$ such that: $f(-|x)$ is a probability measure on $Y$ for ...
Tobias Fritz's user avatar
  • 5,775
1 vote
1 answer
83 views

Intersection of (relativized/preimage) measure 0 with every hyperarithmetic perfect set

Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual ...
Peter Gerdes's user avatar
  • 2,551
4 votes
1 answer
493 views

Complexity of |a| < |b| for ordinal notations?

What is the complexity (e.g. is it $\Sigma^0_1$, arithmetic, fully $\Pi^1_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)? What about the case where only one ...
Peter Gerdes's user avatar
  • 2,551
3 votes
1 answer
180 views

Competing definitions of smooth orbit equivalence relation

Suppose that $X$ is a standard Borel space (meaning it is endowed with a $\sigma$-algebra coming from some Polish topology on $X$) and $G$ is a Polish group acting in a Borel way on $X$. Denote by $...
Iian Smythe's user avatar
  • 2,991
10 votes
0 answers
310 views

Determinacy coincidence at $\omega_1$: is CH needed?

This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
Noah Schweber's user avatar
2 votes
1 answer
106 views

$\Pi^0_2$ singleton forming minimal pair with $0''$

Is there a $\Pi^0_2$ singleton that forms a minimal pair with $0''$? That is, is there a set $X$ such that $X$ is the unique solution to $\forall x \exists y \phi(X|_y, x)$, $X$ and $0''$ are ...
Peter Gerdes's user avatar
  • 2,551

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