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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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A Baire space with meager projections

Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
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Is a Borel image of a Polish space analytic?

A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$. We say that a topological ...
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Countable network vs countable Borel network

Definition. A family $\mathcal N$ of subsets of a topological space $X$ is called $\bullet$ a network if for any open set $U\subset X$ and point $x\in U$ there exists a set $N\in\mathcal N$ ...
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Arithmetic sums of Marczewski null sets

First, recall the Marczewski ideal, called $s^0$: a set $A$ of reals is in $s^0$ iff for every perfect set $P$ there is some perfect $P' \subset P$ such that $P' \cap A = \emptyset$. Secondly, by way ...
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264 views

A strong plus-one hypothesis

To make this more easily readable, I'll start with the question and then give the explanation/motivation. Question. Is the following principle (or its weakening, with "for every real $r$" replaced ...
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1answer
326 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
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1answer
362 views

reverse mathematics of the Lebesgue measurability of analytic sets

Can the fact that all analytic sets are Lebesgue measurable be proven in $Z_2$, or in some weak subsystem such as $\Pi^1_1\text{-CA}_0$? Conversely, can certain set existence axioms be derived from ...
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359 views

Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?

Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...
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1answer
140 views

Support of a regular measure Reg

Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...
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1answer
291 views

$\Delta^{1}_{2}$ and degrees of constructibility $\textbf{on sets}$

This is a follow-up to my question on $\Delta^{1}_{2}$ and degrees of constructibility of real numbers that was answered by the user "William", see here: Can $\Delta^{1}_{2}$ separate degrees of ...
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107 views

Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
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Can $\Delta^{1}_{2}$ separate degrees of constructibility?

Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the ...
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1answer
234 views

Can we inductively define Wadge-well-foundedness?

For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...
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Separation principle in generic models?

Separation principle $\mathbf{Sep}(\mathbf\Sigma^1_n,\mathbf\Delta^1_n)$ claims that any two disjoint pointsets of boldface class $\mathbf\Sigma^1_n$ are separated by a $\mathbf\Delta^1_n$ set. ...
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How much can complexities of bases of a “simple” space vary?

Given a countable subbase of a topology, we can consider its complexity in terms of the difficulty of determining whether one family of basic open sets covers another basic open set. My question is ...
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1answer
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Differences between the reduced Borel field and the category algebra of a space

Let $X$ be a topological space. Halmos calls "reduced Borel field" the quotient $B(X)/M(X)$ where $B(X)$ is the Borel field of $X$ and $M(X)$ is the $\sigma$-ideal of meagre subsets of $X$. Fremlin ...
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Characterizing compact Hausdorff spaces whose all subsets are Borel

I am interested in characterizing compact topological spaces all of whose subsets are Borel. In this respect I have the following Conjecture. For a compact Hausdorff space $X$ the following ...
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1answer
691 views

Interpreting a space in Baire space: how many facts do I need to understand the whole thing?

Below I'm working in ZF+DC+AD or similar; I want enough choice that things don't explode, but I also want the Wadge hierarchy to be well-behaved everywhere. Since this question is a bit long, I've put ...
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1answer
368 views

When can I “draw” a topology in Baire space?

The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free ...
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1answer
117 views

Topologically Ordered Families of Disjoint Cantor Sets in $I$?

Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
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Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?

Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$? Remark 1. The answer to the ...
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Co-cones in the Turing degrees

Let the cocone of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $...
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A connected Borel subgroup of the plane

It is known that the complex plane $\mathbb C$ contain dense connected (additive) subgroups with dense complement but each dense path-connected subgroup of $\mathbb C$ necessarily coincides with $\...
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1answer
83 views

A commuting pair of functions with no common bijective part

Let $X$ be set. Let $f$ be a function from $X$ into $X$. For a given set $E\subseteq X$, we say $E$ determines a $U$-part of $f$ if $f(E)\subseteq E$ and the restriction $f:E\to E$ is a bijection. ...
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1answer
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Closed balls vs closure of open balls

We work in a separable metric space $(X,d)$. With $\overline{B}(x,r)$ I denote the closed ball around $x$ of radius $r$, and with $cl \ B(x,r)$ I denote the closure of the open ball. Clearly, we ...
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How (non-)computable is set theory?

Here is a naive outsiders perspective on set theory: A typical set-theoretical result involves constructing new models of set theory from given ones (typically with different theories for the original ...
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1answer
111 views

Nice representation of open sets in $\sigma$-algebras in certain circumstances

Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Assume ...
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1answer
84 views

Nice arrangement of open sets in $\sigma$-algebras

Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$. Q. Let $O$ be an open ...
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ZF + “every Suslin set of reals is ${\bf \Sigma}^1_2$”

What is known about the theory ($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"? By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...
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Borel class of multivalued mapping

I'm researching Borel class of mapping $G: 2^Y \mapsto 2^X$, where $X,Y - $ compact metric spaces, $f: X \mapsto Y$ $-$ 1st class Borel mapping and $G(A)=cl(f^{-1}(A))$ for $A \in 2^Y$. My idea of ...
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An example of a Borel map of the first class

Let $X,Y$ be compact metric spaces, $2^X$ the set of all closed subsets of $X$ and $f:X\to Y$ be a 1st class Borel mapping. Im trying to check Borel class of mapping $G:2^Y\to 2^X$. I submit it in a ...
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1answer
201 views

Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$

Definitions: Let $X$ be a Polish space (separable completely metrizable topological space). A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...
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1answer
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Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$

We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions. One can generalize the definition above by taking pointwise limit of ...
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Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals: $\mathfrak p$ is the ...
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1answer
192 views

Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal. Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$ ...
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Class of Borel mapping of multivalued map

At the students scientific conference I seen paper where were this propositions: Let $X$, $Y$ $-$ compact metric spaces, $2^X$ $-$ the set of all closed subsets of $X$. Proposition 1. Let $f:X→Y$ ...
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1answer
194 views

Automorphisms of power set lattice mod finite

Let $N$ be a countably infinite set and let $\mathcal P$ denote power set. I get that the automorphisms of $(\mathcal P(N),\subseteq)$ are all induced by permutations of $N$. But what can be said ...
3
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1answer
201 views

Measurably-isomorphic subsets of polish spaces and the continuum hypothesis

In Theorem 2.7 in the following notes, we seem to assume the following statement. Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
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1answer
358 views

The Axiom of Determinacy and the Banach-Mazur game

The Wikipedia article on the Axiom of Determinacy (AD) claims: Equivalent to the axiom of determinacy is the statement that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is ...
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1answer
140 views

A version of the Martin–Solovay tree for $H_\kappa$

Consider a fixed $\Pi^1_2$ property of reals, $A(x)$. Is it true that relative to a regular cardinal $\kappa$, one can define a version of the Martin–Solovay tree $T_2$ for $A$ with the ...
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1answer
230 views

Countable $\mathbf\Sigma^1_2$ equivalence relations

This question is meant to be viewed under moderate large cardinal hypotheses, e.g., enough to ensure $\aleph_1^{L[x]}<\aleph_1$ for all reals $x$. In analogy with the (well-developed) theory of ...
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A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire, if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
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242 views

A strange planar set and the Continuum Hypothesis

Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...
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1answer
169 views

Is there a first-countable space containing a closed discrete subset which is not $G_\delta$?

Being motivated by this problem, I am searching for an example of a first-countable regular topological space $X$ containing a closed discrete subset $D$, which is not $G_\delta$ in $X$. It is easy ...
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1answer
346 views

Is a locally finite union of $G_\delta$-sets a $G_\delta$-set?

Problem. Let $\mathcal F$ be a locally finite (or even discrete) family of (closed) $G_\delta$-sets in a topological space $X$. Is the union $\cup\mathcal F$ a $G_\delta$-set in $X$? Remark. The ...
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1answer
262 views

Equivalent of Lusin's Theorem in Borel setting

Let $X$ be a Polish space, $\mathcal B$ the sigma-algebra of Borel sets. Let $E$ be an aperiodic countable Borel equivalence relation on $X \times X$ (this means that every class of equivalence ...
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147 views

The Fubini property of the $\sigma$-ideal generated by closed subsets of measure zero

Question. Can a Borel subset $B\subset\mathbb R\times \mathbb R$ be covered by countably many closed sets of measure zero in the plane if for every $(a,b)\in\mathbb R\times\mathbb R$ the horisontal ...
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1answer
409 views

Can the Turing degrees be linearly ordered?

Assuming the axiom of choice, every set can be linearly (indeed, well-) ordered. However, without choice this can fail, as witnessed most drastically by the consistency of amorphous sets. More ...
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1answer
206 views

On the topological complexity of the set of winning strategies for Gale-Stewart Games

Given a set $A \subseteq \omega^\omega$, let $G_A$ denote the Gale-Stewart game with payoff set $A$ (so player $I$ wants the real built over the course of play to be in $A$ and player $II$ wants it ...
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82 views

Is this concrete set generically Haar-null?

This question is related to this problem on MO about generically Haar-null sets in locally compact Polish groups but is more concrete. First we recall the definition of a generically Haar-null set in ...