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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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Combinations of Lebesgue measurability, the property of Baire and the perfect set property

Lebesgue measurability (LM), the property of Baire (BP) and the perfect set property (PSP) are probably the most prominent among all the regularity properties of sets of reals. Such a set can either ...
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Why use transfinite induction?

Why use transfinite induction to prove? I think the inclusion relation is trivial by transfinite recursion. $\mathbf{11.B}$ The Borel Hierarchy Assume now that $X$ is metrizable, so that every ...
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1answer
130 views

Analytic sets are Lebesgue measurable

Analytic subsets of $\mathbb{R}$ are projections of Borel sets in $\mathbb{R}^2$. I'm trying to understand a proof that these sets are always Lebesgue measurable. One can first prove that analytic ...
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32 views

Vector space of real valued function with size continuum

I know how I can construct a function such that $f^{-1} (y)$ has size less than continuum actually countable. Here is the proof, Define a relationship as following $$x\sim y \ \text{iff} \ x-y\...
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40 views

A continuum-sized convenient category of topological spaces

From the concluding section of Quasi-Polish Spaces by Matthew de Brecht: "It turns out that the category of quasi-Polish spaces and continuous functions has a very natural description: up to ...
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33 views

Almost clopen sets and Baire property?

A set $A$ is said to have the Baire property if it differs from an open set by a meagre set, that is, if there exists an open set $G$ such that $A\Delta G$ is meager (where $\Delta$ denotes the ...
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1answer
39 views

Given the hierarchy of Borel sets, how to prove that ${\bf \Sigma}_\alpha^0 \subsetneq {\bf \Sigma}_{\alpha+1}^0$ for all ordinal $\alpha<\omega_1$

My textbook Introduction to Set Theory 3rd by Hrbacek and Jech introduces $\mathcal B$, the set of all Borel sets, and then mentions that I have tried, but only to successfully prove that there ...
3
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1answer
61 views

What uniquely characterizes equivalence classes of eventually equal binary sequences?

Let $X$ be the set of all infinite binary sequences. (Or we can think of them as subsets of $\mathbb{N}$ or real numbers between $0$ and $1$.) Let us define an equivalence relation $\sim$ on $X$ by ...
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1answer
192 views

When does a subset of a Polish space meet all the orbits?

While I was studying Borel actions of Polish groups on Polish spaces (I assume of measure $1$), I have tried to understand if a measure $1$ (hence dense) subset of this Polish space meets all the ...
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66 views

How to Modify a Borel function in a Borel way-Self study

I have the following questions: assume that $X$ be a standard Borel space (i.e. a Polish space equipped with the $\sigma$-algebra generated by open sets) with a (possibly Borel, invariant, ergodic) ...
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Almost clopen sets

A set $A$ is said to be almost clopen if it can be uniquely represented in the form $A=B\Delta M$, where $B$ is a clopen set, $M$ is a set of first category (or meagre), and $\Delta$ denotes the ...
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Is every $\sigma$-algebra a Borel $\sigma$-algebra?

Let $\mathscr B$ be a $\sigma$-algebra on a set $X$. Does there always exist a topology $\tau$ on $X$ such that $\mathscr B$ is the Borel $\sigma$-algebra with respect to $\tau$, that is, $\mathscr B$ ...
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113 views

Direct sum of subgroups of $\mathbb{Q}^n$ with $\mathbb{Q}$ is a Borel map - Self study

Let $\mathcal{Pow}(\mathbb{Q}^n)$ be the power set of $\mathbb{Q}^n$ and consider the product topology induced by the natural bijection $\mathcal{Pow}(\mathbb{Q}^n)\cong 2^{\mathbb{Q}^n}$ defined by $...
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1answer
89 views

A consequence of the Selection Theorem for the Effros Borel space F(X) - self study

In Kechris' textbook "Classical Descriptive Set Theory", the following exercise is stated (pp. $77$, Exercise $(12.14)$): "Let X be a measurable space and Y a Polish space. Show that a function $f\...
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1answer
67 views

Borel Hierarchy

i'm in trouble with an exercise on Kechris, Classical Descriptive Set Theory. The Theorem 22.4 shows $\Sigma_\xi^0(X)\neq\Pi_\xi^0(X)$ for each ordinal $\xi\lneq\omega_1$ and uncountable polish space $...
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2answers
76 views

Which classes of subsets are absolute under forcing?

Let $X$ be a ‘definable’ Polish space (in the day-to-day, not necessarily the set-theoretic sense, though possible the latter generalises this). Consider a complexity class $\Gamma(X)$ of subsets of $...
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1answer
37 views

Player II has a winning strategy in the *-game, G*(A), iff A is countable

I have a question concerning the proof of this theorem in Kechris' "Classical Descriptive Set Theory." The rules of the game i as follows: Let $X$ be a nonempty perfect Polish space with compatible ...
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2answers
82 views

Is there a dense set in $\mathbb{R}$ with outer measure $1$? And what about with the Density Topology?

I'm looking for a dense set in $\mathbb{R}$ with outer measure $1$. There is an example like that, but in $[0,1]$ Vitali set of outer-measure exactly $1$.. Also I've tried with Bernstein sets but I ...
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1answer
62 views

Every perfect set has cardinality $2^{\aleph_0}$?

It is well known that perfect sets in $\mathbb{R}^n$ are uncountable (e.g., baby Rudin states this). Recently I heard of this stronger result: Every perfect set in $\mathbb{R}^n$ has cardinality $2^...
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1answer
89 views

Continuous image of a Polish space to another has the Baire property

Here's a theorem in the course notes of a course on Polish groups: Let $X$ and $Y$ be Polish spaces and $f:X\to Y$ continuous. $f(X)$ has the Baire property. In the course note, it's written ...
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1answer
60 views

Power set of Cantor set has cardinality greater than that of reals

I am reading the example of "Boreal measure is not complete" from Wiki: https://en.wikipedia.org/wiki/Complete_measure In the first example, it says The power set of the Cantor set has ...
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63 views

Function with the property $f\circ f=-id$ [duplicate]

Consider a functions $f:\mathbb R\to \mathbb R$ such that $f(f(x))=-x$ for all $x$. It is shown in ``f(f(x)) = − x, Windmills, and Beyond" by Martin Griffiths which appeared in Mathematics ...
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Problem 2. A comprehensive course in Analysis. Barry simon. Page 239.

Definition (Baire set) Let X be a compact Hausdorff space. The Baire sets are the smallest $\sigma$-algebra containing all compacts $G_{\delta}$'s. Definition (Partition) Given an algebra , $\...
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How to show that $g:2^M\to 2^\mathbb{N}$ defined by $g(A) = X\cup A$ is continuous?

In Galvin and Prikry's paper, they inroduce completely Ramsey sets. Definition $5$: A set $S\subseteq 2^\mathbb{N}$ is completely Ramsey if $f^{-1}(S)$ is Ramsey for every continuous mapping $f:2^\...
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1answer
53 views

Are there inner and outer approximation theorems for arbitrary measures?

The outer approximation theorem states that if $E$ is a Lebesgue measurable, then there exists a $G_\delta$ set $G$ containing $E$ such that the Lebesgue measure of $G$ equals the Lebesgue measure of $...
2
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1answer
44 views

Is the projection of a Jordan measurable set Jordan measurable?

The projection of a Borel set need not be a Borel set, and the projection of a Lebesgue measurable set need not be Lebesgue measurable. My question is, is the projection of a Jordan measurable set ...
0
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1answer
76 views

Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

This is a follow-up to my question here. The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be ...
2
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2answers
67 views

Is every Borel subset of a measurable set measurable?

Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$. Let us consider the subspace topology on $A$, and let us consider the Borel sigma algebra under that topology. My question is, is every Borel ...
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1answer
49 views

A dense measure $0$ $G_\delta$ subset of the Fat Cantor set?

The fat Cantor set is a nowhere dense subset of $\mathbb{R}$ with positive Lebesgue measure. My question is, does there exist a $G_\delta$ set dense in the fat Cantor set with Lebesgue measure $0$? ...
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1answer
78 views

A perfect nowhere dense set which intersects every open set with positive measure?

A perfect set is a closed set with no isolated points. A nowhere dense set is a set whose closure has empty interior. My question is, what is an example of a nonempty perfect nowhere dense subset of ...
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1answer
77 views

Why are there continuum many nowhere dense subsets of $\mathbb R$?

I am able to see why the closed nowhere dense subsets of $\mathbb R$ are equinumerous with $\mathbb R$: Every closed nowhere dense subset of $\mathbb R$ is the boundary of an open set (namely its ...
3
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1answer
101 views

Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null $F_\sigma$ set?

The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be written as a union of a Borel set and a subset ...
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1answer
117 views

What is the Sigma Algebra generated by Jordan measurable sets?

Unlike Lebesgue measurable sets, Jordan measurable sets do not form a Sigma algebra. So my question is, what is the Sigma algebra $J$ generated by Jordan measurable sets? All intervals are Jordan ...
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Example of a non-Borel set on Euclidean space

Can someone give an example of a subset of $\mathbb{R}^n$, $n >1$, that is not a Borel set?
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3answers
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Example of a Lebesgue measurable set that can't be constructed from Borel sets and projections?

The Borel sigma algebra on $\mathbb{R}^n$ is obtained by starting with open sets and repeatedly applying the operations of complement, countable union, countable intersection. Now Henri Lebesgue ...
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2answers
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An undetermined measurable set and Borel Determinacy

The existence of a measurable set which is not determined can be proved using the Axiom of Choice (or even without it). Also, we know that $\{ \text{Borel sets} \} \subset \{ \text{measurable sets} \}...
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What sets are obtained by adding $\aleph_1$ unions and intersections to the Borel algebra?

The Borel sigma algebra on $\mathbb{R}$ is obtained by starting with open sets in $\mathbb{R}$ and repeatedly applying the operations of complement, countable union, and countable intersections. My ...
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1answer
117 views

Automorphisms on $(\mathbb R,+)$ and the Axiom of Choice

We know that the algebraic automorphisms of the real numbers under addition is not in $\text{1:1}$ correpondence with $\mathbb R \setminus \{0\}$; see here. The argument uses the AOC. Suppose we ...
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1answer
40 views

Analytic sets have perfect set property (Kechris)

As title says. I’m trying to learn some descriptive set theory but I don’t quite see this. I want to use the following: Given $X, Y$ Polish spaces, $f:X\to Y$ continuous, if $f(X)$ is uncountable ...
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1answer
49 views

Are Baire class functions closed under pointwise limits?

I am confused about the notion of Baire functions (real or complex valued) on a compact space $X$. The set of Borel functions on $X$, $Bo(X)$ is defined to be the set of those functions $f$ for ...
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2answers
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Is the image of a $G_\delta$ set under a continuous mapping of $\mathbb R^n$ a Borel set?

If $A\subset \mathbb R^n$ is a $G_\delta$ set and $f\colon \mathbb R^n\to \mathbb R^n$ is continuous, does it follow that $f(A)$ is a Borel set? It is well-known that the image of a Borel set in $\...
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1answer
48 views

Why doesn't this construction make the rational numbers a $G_\delta$ set?

A $G_\delta$ set is a countable intersection of open sets. The set of rational numbers is not a $G_\delta$ set. But I'm wondering why the following construction doesn't make it a $G_\delta$ set. ...
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1answer
102 views

The Baire space as an automorphism group?

Wikipedia claims that the Baire space "is [the] automorphism group of [a] countably infinite saturated model $\mathfrak{M}$ of some complete theory $T$", however I see neither an obvious group ...
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1answer
166 views

Are “most” sets in $\mathbb R$ neither open nor closed?

It seems intuitive to believe that most subsets of $\mathbb R$ are neither open nor closed. For instance, if we consider the collection of all (open, closed, half-closed/open) intervals, then one ...
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Unique witness form for $\Sigma^1_{n+1}$ formulas

I am interested in equivalently transforming $\Sigma^1_{n+1}$ formulas $\exists Y\;\psi(X,Y)$ (with $\psi$ being $\Pi^1_n$) in the form $\exists Y\;\psi'(X,Y)$, where $\psi'$ is $\Pi^1_n$ and $\psi'(X,...
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1answer
119 views

What is the point of completing a Boolean algebra?

Let $X$ be the Cantor space $2^\omega$ and $Clop(X)$ be the field of open-closed subsets of $X$. It is known that $Clop(X)$ is incomplete, and a minimal completion of $Clop(X)$ is e.g. the regular ...
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1answer
133 views

Can the Interval be Covered by Disjoint Cantor Sets?

I looked in a couple of books focusing on set theory in the real line, but I have not seen the proof or disproof of the following question. It might be something easy just eluding me, but I wonder if ...
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43 views

Can generating set of Borel structure form a basis for Topology.

Let $(X, \mathcal{B})$ be a standard Borel space. Assume we are able to find a sequence of Borel sets $\{A_n\}$ such that $\{A_n\}$ generates the Borel structure. My question is will $\{A_n\}$ also be ...
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1answer
162 views

Is countable choice enough to prove that there are continuum-many Borel sets?

As far as I understand, it is consistent with ZF that every set of real numbers is a Borel set. Furthermore, I know that relatively weak forms of the axiom of choice suffice to prove that $|\mathcal B|...
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3answers
112 views

Can an uncountable subset of $\mathbb{R}$ have empty intersection with its derived set? [duplicate]

Question: Is there an uncountable subset $A$ of $\mathbb{R}$ such that $A\cap A'=\emptyset$, where $A'$ denotes the derived set of $A$? I just know that an uncountable set $A $ must have limit points ...