# 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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### If $f=h\circ g$, then there's a measurable function $\tilde h$ such that $f=\tilde h\circ g$

Let $X,Y,Z$ be three standard measurable spaces and $f:X\to Z$ and $g:X\to Y$ two measurable functions. Suppose that there's a function $h:Y\to Z$ such that $f=h\circ g$. How can I show that there's a ...

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### Periodicity in the cumulative hierarchy

Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the ...

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### Examples of concrete games to apply Borel determinacy to

I'm teaching a course on various mathematical aspects of games, and I'd like to find some examples to illustrate Borel determinacy. Open or closed determinacy is easy to motivate because it proves ...

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### Sufficient condition for the graph of a measurable map to be measurable

Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.
If $(X,\Sigma_X)$ is a standard Borel space can we always ...

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### Consistency of definability beyond P(Ord) in ZF

Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that ...

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### Uniformization and functions on Turing degrees

Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing ...

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### meaning of "the singletons of $\Gamma$ are a basis for $\Gamma$"

In a set of notes for the Cabal Seminar a theorem is attributed to Moschovakis which includes the statement that for certain pointclasses $\Gamma$ the singletons of $\Gamma$ are a basis for $\Gamma$. ...

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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 $\sum a_n$ ...

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### 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 ?

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### 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\...

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### 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 ...

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### 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-...

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### 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(\...

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### 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 ...

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### 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 ...

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### 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) ...

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### 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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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: \...

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### 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 $...

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### 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 ...

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### 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 ...

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### "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 ...

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### 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 ...

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### 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, $\Sigma^2_2$, $\diamondsuit_G$,...

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### 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/....

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### 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 ...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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) ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...

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### 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 ...

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### $\sigma$-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 \...

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### 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.
...

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### 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 $...

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### 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 ...

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### 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 ...