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.

Filter by
Sorted by
Tagged with
3
votes
0answers
38 views

Borel isomorphisms between perfect Polish spaces and their “rank”

Let $\mathcal{X, Y}$ be perfect Polish spaces. Define $\Sigma_1^0$ sets to be open sets, and for $\xi>1$, $\Sigma_\xi^0$ sets to be sets of the form $\bigcup_{n=1}^\infty P_n^c$ where $P_n$ is a $\...
1
vote
1answer
39 views

Wijsman topology

I'm reading a paper and I have a doubt about about a part of it. In the paper it is defined a topology over $\mathcal{F}(X)$ the set of all closed sets of a polish space $X$, which is named Wijsman ...
3
votes
1answer
41 views

Presentation on game theory and determinacy

I have to write a short paper(about 20 pages) and prepare a presentation (about 1 hour) for an exam on Game Theory (it is a general, introductory course). I've looked up some things on the internet ...
8
votes
0answers
112 views

Not quite cardinal characteristics of the continuum

We work in $\mathsf{ZFC}$: Given an ideal $\mathcal{I}$ of subsets of $\omega^\omega$, let $\kappa_\mathcal{I}$ be the smallest cardinality of a collection $\mathfrak{C}$ of $\mathcal{I}$-positive (= ...
7
votes
0answers
93 views

In an uncountable Polish space there is no countably generated $\sigma$-algebra between analytic sets and sets with the Baire property

Let $X$ be an uncountable Polish space. Assume that $\mathcal{A}$ is a $\sigma$-algebra of subsets of $X$ such that every analytic subset of $X$ belongs to $\mathcal{A}$ and every member of $\mathcal{...
2
votes
0answers
70 views

$f: X\rightarrow Y$ s.t. $f^{-1}(\boldsymbol{\Sigma_3^0})\subseteq\boldsymbol{\Sigma_2^0}$

I'd like to prove or disprove (but I think it's true, maybe it is even trivial but I don't see it) the following statement: Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$ then $...
-1
votes
1answer
69 views

Meager sets and the property of Baire [closed]

Suppose that $A$ and $B$ are sets with Baire property, and $U$ and $V$ are open sets, $A\div U$ and $B\div V$ are both meager. Prove that if $A \cap B$ is meager, then $U \cap V = \emptyset$.
0
votes
0answers
32 views

property of Baire and Borel sets

I know that every Borel set has the property of Baire. I know also that the opposite theorem is not true - there exists sets which have the property of Baire ad they are not Borel sets. What is an ...
0
votes
2answers
60 views

Every point of a perfect set in complete metric space is condensation point

here is the one of the answer my question in the cite but I would like to ask something, 'If $(X,d)$ is a complete metric space, and $A$ is perfect and $x \in A$ it follows that for any $r>0$, $D(x,...
0
votes
1answer
102 views

Why is this function, used to show that finite Borel relations are smooth, Borel?

I'm reading a proof that all finite equivalence relations are smooth, which goes as follows: Let $E$ be a finite Borel equivalence relation on a Polish space, we want to find a Borel function $f\colon ...
1
vote
1answer
36 views

Open subset of a polish space

HERE is the theorem about being polish of open subsets of polish spaces. Let $X$ be a Polish space and the complete metric on $X$ is $d<1$ and the new one defined on any open set $U\subset X$ is ...
1
vote
1answer
41 views

Is taking fibers continuous?

Let $f\colon X\to Y$, with $Y$ at least $T_1$, be continuous and consider the hyperspace $F(X)$ of closed sets of $X$ with the Fell topology, which has as basis the sets $$\langle K;U_1,\ldots,U_n\...
1
vote
0answers
42 views

On Power Set of Naturals and Cantor Space

Let $A\subseteq\mathcal{P}(\mathbb{N})$ be any subset of naturals, the charasteristic function of A is for each $i\in\mathbb{N}$ $$\chi_{A}(i) = \begin{cases} 0 & i \notin A \\ 1 & i \in A \...
2
votes
1answer
37 views

Incomparable Cardinal Invariants of the Continuum

For most cardinal invariants of the continuum we know one to always be greater or equal to the other. Consider for example $\mathfrak p \leq \mathfrak t \leq \mathfrak b$ which is somewhat elementary. ...
1
vote
1answer
28 views

Borel space of dynamical subsystems

Let $ T $ be an invertible function acting by homeomorphisms on a compact Polish space $ X $ (i.e., a continuous action of $ \mathbb{Z} $). Denote by $ [x] $ the orbit of $ x $ under $ T $: $ [x] = \{\...
0
votes
1answer
53 views

Prove $C$ and $S$ are $\Sigma^1_1$ and $\Pi^1_1$ respectively.

Let $({}^{\omega_1}\omega_1,\tau)$ be a topological space where $\tau$ is generated by $\{N(f,\alpha)\mid f\in{}^{\omega_1}\omega_1, \alpha<\omega_1\}$ and $N(f,\alpha)=\{g:\omega_1\to\omega_1\mid ...
0
votes
2answers
41 views

Baire space is homeomorphic to countably many copies of itself

On wikipedia I found that the Baire space $\mathcal{N}$ is homeomorphic to the product of a countable number of copies of itself, however, I haven't been able to find a proof. The Baire space is ...
0
votes
1answer
45 views

metrics of Product space

Let for each $i\in\mathbb{N}$ $(X_{i},\rho_{i})$ be discrete metric spaces, the metrics defined on product space $\prod_{i\in\mathbb{N}}X_{i}$ \ $d(x,y)=\sum_{i=1}^{\infty}\frac{\rho_{i}(x_{i},y_{i})}{...
5
votes
1answer
101 views

Exercise on $\boldsymbol{\Delta}_2^0$ sets in the Baire space $\omega^\omega$

I want to prove the following statement: Given a set $A\subseteq \omega^\omega$, if there exists a continuous function $g: 2^\omega \rightarrow \omega^\omega$ s.t. $f(\{z \in 2^\omega \mid \exists n \...
0
votes
1answer
28 views

Countable dense subset of Baire (ω^ω)

Prove that $$ D = \{ f \in \mathbb{N}^\mathbb{N} \mid \exists i \forall j \geq i f(j) = 0 \} $$ is a countable dense subset of $\mathbb{N}^\mathbb{N}$ (Baire). Here is the theorem about the ...
4
votes
1answer
157 views

About the transference of perfect sets under Polish group actions

Let $G$ be a Polish group, $X$ a Polsih space on which $G$ acts continously and consider the orbit equivalence relation on $X$ with respect to $G$. Suppose $A\subseteq X$ is a perfect set of pairwise ...
1
vote
3answers
32 views

Characterizing elements of a collection closed under countable intersections

Suppose we have a collection $\mathcal{E}$ containing some subsets of a nonempty set $E$. Assume $E \in \mathcal{E}$. Define $\hat{\mathcal{E}}$ to be the smallest collection containing elements of $\...
2
votes
1answer
63 views

Why do we have $E_0(\Bbb N^\Bbb N)\sim_c E_0$?

Let $E_0$ and $E_0(\Bbb N^\Bbb N)$ be the relations of eventual agreement on $2^\Bbb N$ and $\Bbb N^\Bbb N$, so $$xE_0y\iff\exists m\forall n\geq m x(n)=y(n)$$ and similarly for $E_0(\Bbb N^\Bbb N)$. ...
0
votes
1answer
31 views

Baire space is zero dimensional

Here is the basis of Baire's topology with $s\in A^{<\mathbb{N}}$ and $\mathcal{B}=\{N_{s}:s\in A^{<\mathbb{N}}\}$ I was trying to show that elements of $\mathcal{B}$ is both open and closed. I ...
2
votes
1answer
31 views

On gaps in the dimension of subspaces

Let $P\subseteq\Bbb N\cup\{\infty\}$, with $0\in P$. Does there exist a "nice" space $X_P$ such that $X_P$ has a subspace of dimension $n$ iff $n\in P$? By nice I mean nice enough that there'...
2
votes
1answer
65 views

Uniformization property of $\mathbf{\Sigma}^0_n(\mathscr X\times\omega)$ for $n>1$

This question is about exercise 1C.6 from Moschovakis. The exercise is to prove if $P\subset\mathscr X\times \omega$ is in $\mathbf{\Sigma}^0_n$ for $n>1$, then there is some $P^*\subset P$ that is ...
1
vote
0answers
27 views

Models for which Measurable implies Universally Baire?

To begin, Universally Baire sets are Lebesgue measurable (slide 4 of http://www.ub.edu/RSTR2018/slides/Woodin.pdf). I was wondering if there are any natural models of ZFC where the converse holds, and ...
6
votes
0answers
63 views

On positive dimensional Polish spaces in which every compact set has empty interior

A standard characterization of the Baire space is that is the only nonempty, zero dimensional, Polish space in which every compact set has empty interior (up to homeomorphism of course). I'm ...
0
votes
1answer
55 views

Every compact subset of Baire Space (ω^ω) has empty interior [duplicate]

Here is the Well known proposition. I assumed that K is the compact subset of ω^ω and it contains an open set Ns, so I thought if K is compact so Ns is because Ns is clopen set, If I can not find a ...
-2
votes
2answers
87 views

Every compact subset of Baire space $\omega^\omega$ has empty interior. [duplicate]

Here is the well-known proposition, I assumed that $K$ is compact subset of $\omega^\omega$ and it contains an open Set $N_s$, so $N_s$ is clopen and thus it is compact. If I can show $N_s$ can not be ...
5
votes
1answer
55 views

Does the family of compact subsets of a Polish space belong to the Effros Borel structure?

Since the answer to my previous question (see Effros Borel structure ) turned out to be negative, I don't know how to solve the following problem: Let $X$ be a Polish space and let $\mathcal{F}(X)$ be ...
1
vote
1answer
67 views

Effros Borel structure

Let $X$ be a Polish space and denote by $\mathcal{F}(X)$ the set of all closed subsets of $X$. The Effros Borel structure on $\mathcal{F}(X)$ is the smallest $\sigma$-algebra on $\mathcal{F}(X)$ ...
2
votes
1answer
51 views

Elementary confusion about determinacy

In ZF we can prove Borel determinacy. In ZF we can prove that the determinacy for closed sets is equivalent to the Axiom of Choice. Since closed sets are Borel, it follows that the Axiom of Choice is ...
0
votes
1answer
68 views

Homeomorphic images of endpoints of the Cantor set.

A Cantor set is a homeomorphic image of the standard ternary Cantor set $T$. Suppose that we have a Cantor set $C$ on the plane. It is well know that $C$ is in fact ambiently homeomorphic to $T$, ...
2
votes
0answers
37 views

The supremum of uncountably many upper semmianalytic functions

I've been trying to learn a bit about analytic sets, and upper/lower semianalytic functions through the book, Stochastic optimal control, the discrete-time case. In chapter 7, Lemma 7.30, it said that ...
0
votes
1answer
22 views

Iteratively constructing Borel sets

From this post, given a collection of sets $\mathcal{C}_0$, we can construct $\sigma(\mathcal{C}_0)$ through an iterative process, where for each step we let $\mathcal{C}_\alpha$ be the collection of ...
0
votes
1answer
43 views

Complexity of a satisfaction relation

Given $x, y \in \omega^\omega$, we may define an expanded structure $(\mathbb{N}, x, y)$ where $x,y$ are treated as interpretations of function symbols. I want to show that there is a computable set $...
1
vote
1answer
27 views

Finer Polish topology which turns countably many given Borel sets to clopen sets

Let $(X,\tau)$ be a Polish space and $(B_n)_{n \in \omega}$ a sequence of Borel sets in $(X,\tau)$. I would like to know if this implies that there is a Polish topology ${\tau}_{\omega}$ on $X$ such ...
0
votes
0answers
34 views

Prove that the relation $f'(x)$ exists is ${\bf{\Pi }}_0^3$

I'm having a trouble with 1C.3. in Moschovakis' Descriptive Set Theory. The problem is as follows. Let $f:\mathbb{R} \to \mathbb{R}$ be continuous. We need to show that the relation $$Q(x) \...
0
votes
2answers
55 views

Relationship between meager subsets of $\mathbb{R}$ and $\omega^\omega$.

I'm reading this book about structure of the real line. One of the things that's proven there is that $\mathfrak{b}\leq non(\mathcal{M}).$ And the proof goes as follows: For every $f\in\omega^\omega$ ...
1
vote
2answers
29 views

Souslin operation $\mathcal{\Gamma}$ contains countable union and intersection

This is exercise 25.5 II) in Kechris Descriptive Set Theory: Denoting by $\Gamma_\sigma$, $\Gamma_\delta$ the class of sets that are respectively countable unions or countable intersections of sets in ...
2
votes
1answer
31 views

Polish group actions: if an orbit is non-meager in itself, it is a Baire space?

Assume that $G$ is a Polish group continuously acting on a Polish space $X$. Let $x \in X$ be a point such that $G \cdot x$, the orbit of $x$, is non-meager in its relative topology. I would like to ...
0
votes
0answers
48 views

Construction a Bernstein set that is an additive group

I would like to start by showing that there exists a Bernstein set $B$ that is additive group. To see that, it is an easy transfinite induction to construct a Bernstein set that is a linear ...
0
votes
2answers
37 views

Characterizing open countable subsets of the Cantor set

The Cantor-Bendixson theorem implies that any closed subset of the Cantor set $\mathcal{C}$ can be described as a disjoint union of a set $\mathcal{C}_c$ that is homeomorphic to the original Cantor ...
0
votes
1answer
45 views

Meager set and disjoint with line $y=ax$

Lemma. Let $X$ and $Y$ be second countable. If $A\subset X$ and $B\subset Y$, then $A\times B$ is meager iff at least of $A,B$ is meager. Assume $f\colon \mathbb R\to \mathbb R$ and $D$ be a meager ...
1
vote
1answer
94 views

Meager set and $\mathrm{MA}$

Martin's axiom. Let $\langle\mathbb P,\leq\rangle$ be a ccc partially ordered set. If $\mathcal D$ is a family of dense subset of $\mathbb P$ such that $|\mathcal D|<\mathfrak c$, then there exists ...
4
votes
1answer
36 views

Proof that Effros Borel space is standard

I have difficulty understanding the proof of Theorem 12.6 in Kechris's Classical Descriptive Set Theory that if $X$ is Polish then the Effros Borel space of $F(X)$ is standard. $F(X)$ consists of all ...
1
vote
0answers
61 views

Co-meager contains a perfect set

Let $A\subset\mathbb R$ be a meager set and $I\subset\mathbb R$ be an open interval. So, $I\setminus A$ is co-meager in I. I know, $I\setminus A$ has ccardinality $\mathfrak c$ and dense in $I.$ I ...
1
vote
1answer
26 views

Analytic sets and equivalence relation

Let $X$ be a polish space and $E$ an equivalence relation on $X$ such that $E\subseteq X^2$ is analytic. Show that for every analytic set $A\subseteq X$ the set $[A]_E=\{x\in X: \exists y\in A: xE\,y\}...
2
votes
0answers
46 views

Is an uncountable, compact, totally disconnected subset of a Cantor set also a Cantor set?

A closed subset of the Cantor set with the usual metric is necessarily compact and totally disconnected. However, it is not necessarily perfect—for example, one can always pick a set of two points at ...

1
2 3 4 5
17