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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.

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Is there a function $f :A^{<\mathbb{N}}\to E$ such that $f (s{}^\frown a)=h(f(s),a,|s|)$ for any $s\in A^{<\mathbb{N}}$ and $a\in A$?

Given a set $A$, define $A^{<\mathbb{N}}:=\cup _{n\in\mathbb{N}}A^n$ with $A^0:=\{\emptyset\} $ and $A^n$ being the $n$-fold cartesian product of $A$. For any $s:=(s_0,\cdots,s_{n-1})\in A^n\...
rfloc's user avatar
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What's the intuition behind the definition of "rank of a tree"?

Let $T$ be a well-founded tree on $\mathbb{N}$, that is, the set of infinite branches of $T$ is empty. Define a function $\rho_T : T\to \mathbf{ON}$ inductively by $$\rho_T(u)=\sup\{\rho_T(v)+1: u\...
caligulasremorse's user avatar
2 votes
1 answer
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Complement of any dense countable subset of reals is homeomorphic to irrationals

I recently stumbled upon this: For any infinite countable subset $A\subseteq\mathbb R$ such that $\overline A=\mathbb R$, the complement $\mathbb R\setminus A$ is homeomorphic to the Baire space. (Or,...
Martin Sleziak's user avatar
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A question on an equivalent form of Kalmar hierarchy

In the book named A Course on Borel Sets by Srivastava, the set of all clopen subsets of $\mathbb{N}^\mathbb{N}$ is given as in the following screenshot. However, the book doesn't provide with a ...
boyler's user avatar
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The relation between the cardinality of Bore $\sigma$-algebra and axiom of choice

Under axiom of choice, the cardinality of Borel $\sigma$-algebra $B$ is $\mathfrak{c}$. In this proof axiom of choice is used three times: To prove $\omega_1$-times recursion is sufficient, each $|B_{...
Gizerst Nanari's user avatar
3 votes
1 answer
57 views

Help in understanding this example in Kechris on the Borel heirarchy

I am trying to read Kechris's descriptive set theory (self-study only). In Chapter 22 on the Borel heirarchy, let $A \subseteq \mathbb{Q}$ be such that $A$ and $\mathbb{Q} \setminus A$ are dense (I ...
Link L's user avatar
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A Borel set with convex sections has a Borel projection

The following is an exercise from "A Course on Borel Sets" by S.M. Srivastava. Exercise 4.7.10 Let $X$ be a Polish space and $B \subseteq X \times \mathbb R^n$ a Borel set with convex ...
J.R.'s user avatar
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34 views

Invertibility of a measurable mapping from lower and upperbounds on the induced pushforward measure

Let $\Omega \subseteq \mathbb{R}$ be open and consider the standard Borel space $(\Omega, \mathcal{B}(\Omega), \mu)$, where $\mu$ denotes the Lebeasgue measure. Let $f: \Omega \to \Omega$ be a ...
Saleh's user avatar
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Is the set of all antichains of $\omega^{<\omega}$ avoiding all chains Borel?

Let's identify $\mathcal P(\mathbb N^{<\mathbb N})$ with $2^{\mathbb N^{<\mathbb N}}$, so it is a Polish space. The set $\mathcal A=\{A\subseteq \omega^{<\omega}: A \text{ is a maximal ...
Pink lake's user avatar
3 votes
1 answer
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Is there a sequence of two-set partitions of $[0,1]$ such that the partition generated by any subsequence is a partition into finite sets?

More specifically, I'm wondering if there is a sequence of partitions $\{\mathcal{P_n}\}_{n\in \mathbb{N}}$ of the unit interval $[0, 1]$, where each $\mathcal{P}_n$ consists of precisely two sets, ...
Jonathan Hole's user avatar
10 votes
1 answer
251 views

Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density

Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
confused's user avatar
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Clarification on "absolute" property

I read that if some property $P(x)$ defined by a formula $\phi$ is absolute for some class $M$, then $\phi(P(x)) \leftrightarrow \phi^M(P(x))$, where $\phi(P(x))$ is interpreted in $V$, and $\phi^M(P(...
Link L's user avatar
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Equivalence of definitions of "standard Borel space"

I met the following definition of standard Borel spaces in Durrett's probability theory book (slightly rephrased): $(S,\mathcal{S})$ is said to be standard Borel if it is isomorphic (as a measurable ...
J. Doe's user avatar
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Help in understanding this part of Mansfield / Weitkamp

I am trying to read Effective Descriptive Set Theory by Mansfield and Weitkamp (self-study only so please bear with me ... am a beginner in this area). I need help in understanding Example 1.24 of the ...
Link L's user avatar
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1 answer
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A question about Bartoszynski (Set Theory, On the structure of the real line): Small sets

I have a question about Theorem 4.1.3 in Set Theory, On the structure of the real line. The theorem states that the following conditions are equivalent: (1) $\mathcal{F}$ is measurable. (2) $\mathcal{...
C_M's user avatar
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2 answers
71 views

A countable intersections of derived sets

This question is from Hrbacek and Jech's intro set theory book (Ch 10, exercise 4.6). I am given: $$F=\{1\}\cup\{1-\frac{1}{2^{n_1}}-\frac{1}{2^{n_1+n_2}}-...-\frac{1}{2^{n_1+...+n_k}} : k\leq n_1 \...
sleepysleepy's user avatar
2 votes
0 answers
70 views

Countable union of graphs of continuous functions is closed

As context, I am trying to prove that Borel sets are analytic, i.e. given a Polish space $X$, there is some Polish space $Y$ such that $B \subseteq X$ is Borel and is the image of a continuous ...
Link L's user avatar
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1 vote
1 answer
60 views

Existence of measurable section

Let $P \subset S^{d-1} \times \mathbb{R}^d$ be a compact set such that $proj_{S^{d-1}}(P) = S^{d-1}$, I want to show that there exists a Borel measurable map $ f \, : \, S^{d-1} \mathbb{R}^d$ such ...
Paul's user avatar
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1 answer
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Why is this set $Z$ in $\Pi_n Y_n$ closed

I am trying to read Kechris book on Classical Descriptive Set Theory (self-studying only so please do bear with me). I need help in understanding the following part of the proof for (i) of Proposition ...
Link L's user avatar
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5 votes
2 answers
552 views

Three Notions of Small

I’m currently learning about the history of the development of the Lebesgue integral in Thomas Hawkins’s book “Lebesgue’s Theory of Integration; It’s Origins and Development” Hawkins is stressing how ...
Joe's user avatar
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Definition of a Partial Order (In: Kanamori, The Higher Infinite)

I have a question regarding a definition in Kanamori's book on page 136. (it's the beginning of the subsection about Random Reals) Let $\mathcal{B}^{\star} = \{X \in \mathcal{B} \ \lvert \ X \ \text{...
C_M's user avatar
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1 answer
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Is a totally ordered set consisting of discrete point sets containing point p a well-ordered set?

I have encountered a fully ordered set and want to determine whether it is well ordered. Each element B in this totally ordered set A is a set and includes point p, which means that the elements in A ...
lllka's user avatar
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1 vote
1 answer
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Set of infinite well founded trees if complete coanalytic?

I know that the set $WF$ of well founded trees in $\omega^\omega$ is a coanalytic complete set and I was wondering if I restrict to the set of all inifinite (as sets) trees, the set of all infinite ...
Eparoh's user avatar
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4 votes
0 answers
77 views

Suslin measurable sets and the smallest field containing all analytic sets

Let $X$ be a Polish space. Recall that the Suslin operation is the operation $\mathcal{A}$ such that for any Suslin scheme $\{A_s : s \in \omega^{<\omega}\}$ of subsets of $X$, we have: $$ \mathcal{...
Clement Yung's user avatar
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Why is transfinite recursion needed for the construction of Borel sets? [duplicate]

I'm reading the Hrbaceck and Jech's book and they say that recursion until $\omega$ does not work for constructing the Borel sets But I haven't been able to find such sequences, do you have any ...
Selena's user avatar
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2 answers
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Essentially infinite continuous maps $\prod_{\mathbf{N}} \mathbf{N} \rightarrow \mathbf{N}$?

Let $\mathbf{N}$ be the natural numbers with the discrete topology (or really, any countable set), and consider the space of natural number sequences $\prod_{\mathbf{N}} \mathbf{N}$ with the product ...
TSBH's user avatar
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5 votes
1 answer
146 views

Confused by this proof in Jech's set theory

In Jech's Set Theory, Chapter 11, the universal set $U$ is defined as: For each $\alpha \geq 1$, there exists a set $U \subset \mathcal{N}^2$ such that $U$ is $\Sigma_{\alpha}^0$ (in $\mathcal{N}^2$) ...
Link L's user avatar
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6 votes
1 answer
114 views

$L[a] \cap 2^{\omega}$ is $\Sigma_2^1$

I have the following question: Let $a\in \mathbf{R}$ sucht that $X = L[a] \cap 2^{\omega}$ is uncountable. Why is $X$ is a $\Sigma_2^1$ set? $L[a]$ is the inner model that can be built by ...
Caro Meier's user avatar
1 vote
1 answer
93 views

Theorem of Shelah about the existence of an inaccessible cardinal

There is a theorem of Shelah, stated in the following way: If all $\Sigma_3^1$ sets of reals are measurable, then $\aleph_1$ is an inaccessible cardinal in $L$. In some textbooks (for example ...
C_M's user avatar
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3 votes
0 answers
90 views

Extension of Borel map from a separable metric space to a Polish space

Suppose that $f:X\to Y$ is a Borel map from separable metric space $X$ to a $T_3$ space $Y$. Does there always exist a Polish space $\tilde X \supseteq X$ and $T_3$ space $\tilde Y\supseteq Y$ and an ...
Jakobian's user avatar
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2 votes
1 answer
66 views

Equivalent condition for measurable filters.

It's a known fact (see for example Bartoszynski and Judah: Set Theory, On the Structure of the real Line) that for a filter $F$ on $\omega$ the following conditions are equivalent: $F$ is Lebesgue ...
Caro Meier's user avatar
1 vote
1 answer
62 views

On the right inverse of a Borel measurable map between two Souslin spaces being Souslin measurable, i.e. mapping open sets back to analytic sets

I'm currently reading some stuff on the existence of a measurable inverse, or measurable choice theorems from Bogachev's Measure Theory book, Volume II, where I'm trying to connect these two theorems ...
Learning Math's user avatar
0 votes
0 answers
23 views

Family of intersection compact with open set generates borel of compact sets

Let $X$ Polish space, and $K(X):\{K \subset X : K \text{ is compact}\}$ I need a hint how to prove the family of sets $\{K \in K(X): K \cap U \neq \emptyset\}$ and $\{K \in K(X): K \subset U\}$ ...
PSW's user avatar
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3 votes
0 answers
139 views

Schindler, 2014: 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
1 vote
1 answer
89 views

About the rank of a tree

I've found three different definitions of rank for a well-founded tree and I have some questions about it. First of all, these are the definitions: Definition 1: Let $T$ be a well-founded tree on $\...
Eparoh's user avatar
  • 1,279
7 votes
2 answers
415 views

Does satisfaction at all arithmetical sets of a second-order arithmetic formula with no bound predicate variables imply its satisfaction?

Let $\varphi(X_1,\ldots,X_r)$ be a second-order arithmetic formula with no bound predicate variables and free predicate variables $X_1,\ldots,X_r$ (all of arity $1$ for simplicity). Assume every ...
Gro-Tsen's user avatar
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2 votes
0 answers
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Simpson's proof of the Gandy Kreisel Tait theorem in $\textbf{ATR}_0$

I believe there is an error in Simpson's book "Subsystems of Second Order Arithmetic". Theorem VIII.6.4 states: $\textbf{ATR}_0$ proves that any $Y$ such that $Y_i=\{(n,i)\in y:n\in\mathbb{N}...
Giorgio Genovesi's user avatar
0 votes
1 answer
75 views

Cantor-Bendixson Theorem

I know two different proofs of the Cantor-Bendixson theorem, however both explicitly construct the perfect set and the countable set directly, without using the fact that closed sets have the perfect ...
Niko Gruben's user avatar
1 vote
0 answers
84 views

Proof of Shelah's Theorem about inaccessible cardinals using Raisonnier Filters

I'm studying Chapter 9, titled The Raisonnier Filter in Ralf Schindler's textbook (Set Theory: Exploring Independence and Truth, 2014). The goal of this chapter is to prove Shelah's theorem: Assume ...
C_M's user avatar
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0 votes
0 answers
60 views

A derived set of a specific countable union of sets

For every $n>1$, let $A_n = \{\frac{1}{n}+\frac{1}{kn(n-1)}\ : k \in \mathbb{Z}^{+}\}$. What is $$\left(\bigcup_{n=2}^{\infty} A_n \right)'?$$ My initial thought that it should be $\{0\}\cup\{\frac{...
fade idham's user avatar
3 votes
1 answer
82 views

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

Erdős-Sierpinski 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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1 vote
1 answer
41 views

Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.

If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
Susana Santoyo's user avatar
1 vote
1 answer
47 views

Show that the following metric is complete on $L_1(X,Y)$

I am working through the exercises of Kechris' Classical Descriptive Set Theory for personal edification. I've solved Exercise 3.5, but the proof was two pages. Did I miss a more straightforward ...
Eli Johnson's user avatar
0 votes
1 answer
60 views

Are projective sets from descriptive set theory projective objects in Set category?

There are projective sets in descriptive set theory. For them, the axiom of determinacy is not contradicting the axiom of choice. Given the axiom of choice, every set is a projective object. But in a ...
uhbif19's user avatar
  • 360
6 votes
1 answer
147 views

Why isn't $[0,1]^{\aleph_1}$ isomorphic to $[0,1]$ as a measure space.

I was talking with a professor, and he mentioned that $[0,1]$ (with Lebesgue $\sigma$-algebra and Lebesgue measure) isn't isomorphic as a measure space to $[0,1]^{\aleph_1}$ (with the product measure)....
Susana Santoyo's user avatar
2 votes
1 answer
166 views

Alpha recursion - Constructible universe and Analytical hierarchy

Alpha recursion and Constructible universe are very intertwined, because the first is based on the concept of admissible ordinal $\alpha$ which is defined as an ordinal such that $L_\alpha$ - a set ...
holmes's user avatar
  • 423
1 vote
1 answer
91 views

Two set-theoretical assumptions

My question about the following two set-theoretical assumptions: union of less than continuum many meager subsets of $\Bbb R$ is meager in $\Bbb R.$ Union of less than continuum many meager subsets ...
00GB's user avatar
  • 2,423
2 votes
1 answer
79 views

Given $p\in[0,1]$ form a set $\Omega$ and a function $f$ such that $\{f = 1\} = p$ and $\{f = 0\} = 1-p$

Given $p\in[0,1]$ is it always possible to form a set $\Omega$ and a function $f$ such that the expressions below are true? $$ \begin{align} \frac{|\{\omega\in\Omega\,:\, f(\omega) = 1\}|}{|\Omega|...
Physics_Student's user avatar
1 vote
0 answers
42 views

Diagonal of the power of a standard Borel space is isomorphic to the original space

I don't have much experience with measure spaces, so I am asking for verification of some proofs. Thank you. Let $(X, \Sigma)$ be a standard Borel space. Consider the product space $(Z, \Sigma') = (X, ...
Tomáš Hons's user avatar
3 votes
1 answer
59 views

Dense co-dense subsets of a subspace of the Baire space

Let ${\mathbb{N}^\mathbb{N}}^*$ denote the infinite sequences in Baire space that do not have a periodic tail, which is a $G_\delta$ subset so it is a Polish space. Consider the equivalence relation ...
Onur Bilge's user avatar

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