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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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Can we describe any subsets of $\mathbb{N}$ occurring in a late layer of the Constructible Universe?

There is a certain large countable ordinal referred to in the literature as $\beta_0$. It was first discovered by Paul Cohen, and here are some equivalent characterizations of it: The smallest ...
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Cantor-Bendixson Theorem Proof (existence of an ordinal)

Let $A$ be a topological space, denote the set of accumulation points of $A$ by $A'$. If $\alpha$ is an ordinal, we define $A^{(\alpha)}$ by transfinite induction: $A^{(0)} = A$, $\, \,A^{(\alpha+...
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Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$ [duplicate]

I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is ...
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Intuition of Standard and Analytic Borel space

An analytic Borel space is a countably generated Borel space which is the image of standard Borel space under a Borel mapping. A standard Borel space is the Borel space associated to a Polish space. ...
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Why $IF^*$ is $\Sigma^1_1$-complete

The following is the Exercise 4.2.3 in Srivastava's book A course on Borel sets Let $$N=\{\alpha\in {\mathbb N}^{\mathbb N}:\alpha(i)>0 ~\ for ~\ infinitely ~\ many ~\ i\}$$ Show the set $$...
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Compatibility of a metric with a left-invariant metric on a topological group

My question arised while I was reading Kechris' "Classical Descriptive Set Theory", pp. $58$. At row $14$ of that page, the author states that If $d$ is a left-invariant complete metric on a ...
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Separately continuous implies continuous

I don't understand the proof of the following result in Kechris' "Classical Descriptive Set Theory", pp. $62$ Theorem $(9.14)$ Let $G$ be a group with a topology that is metrizable and Baire, s.t....
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Understanding the proof of Kechris' theorem 3.11

In Alexander Kechris' Classical Descriptive Set Theory, he proves a quite useful theorem (3.11) that I'm using as a vital part of a project I'm doing. However, there's a part of the proof I can't wrap ...
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Uncountable Polish Spaces and Borel Pointclasses

I have to prove that for $\alpha<\omega_1$ and unctbl Polish spaces $$\Sigma_\alpha^0\neq\Sigma_{\alpha+1}^0$$ Of course $\Sigma_\alpha^0\subseteq\Sigma_{\alpha+1}^0$ but I would be grateful for a ...
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Is the usual metric on $\Bbb{N}^\Bbb{N}$ left invariant on $S(\Bbb{N})$?

Let $\Bbb{N}^\Bbb{N}$ be the set of all functions $(x_n\mid n\in\Bbb{N})$ from $\Bbb{N}$ into itself (I identify sequences with their images, as usual). I know this is a metrizable space with ...
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Equivalence relation induced by a group action is an analytic set

We say that $X$ is a standard Borel space iff it is a Polish space equipped with the Borel $\sigma$-algebra. Similarly, a standard Borel group is a Polish group s.t. multiplication and inversion are ...
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Automorphisms group of a structure is a closed subgroup of the permutations over $\mathbb{N}$

Here is the second part of Example $7)$ of Kechris' "Classical Descriptive Set Theory" (pp. $59-60$): More generally, consider a structure $$\mathcal{A}=(A,(R_i)_{i\in I},(f_j)_{j\in J},(c_k)_{k\...
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Does every null set have a superset which is an $F_{\sigma}$ null set?

Let $A$ be a Lebesgue null set in $\mathbb R$. Can we find a set $B$ with the following properties: 1) $A \subset B$ 2) $B$ has measure $0$ 3) $B$ is an $F_{\sigma}$ set (i.e. a countable union of ...
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Independence of the uniform metric from the compatible metric for the codomain

Let $X$ be a compact metrizable space and $Y$ a metrizable space. Denote by $C(X,Y)$ the space of continuous functions from $X$ to $Y$ with the topology induced by the uniform metric $$d_u(f,g)=sup_{x\...
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Extending an homeomorphism to a $G_\delta$ set (comment on @Brian M. Scott's proof)

This question has already been posed and aswered, but I think the proof proposed by @Brian M. Scott can be "simplified". If not, I would like to understand why. NOTE: I ask this question because @...
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Body of a pruned tree is closed

By definition, a tree is a set of finite sequences on some alphabet $A$ (equipped with the discrete topology) which is a lower set w.r.t. the initial segment relation. A tree is said to be pruned iff ...
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What is the set of pointwise limits of polynomials?

The set of pointwise limits of continuous functions from from $\mathbb{R}$ to $\mathbb{R}$ is the set of Baire class 1 functions. My question is, my question is, what is the set of pointwise limits ...
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The canonical open set which is equal to a set with the BP modulo meager sets is regular open (Kechris)

A set $U$ in a topological space $X$ is called regular open iff $U=(\overline{U})^°$. Exercise $(8.30)$ (Kechris, "Classical Descriptive Set Theory") Prove that $$U(A)=\bigcup \{U\,\text{open}\...
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Theorem $(8.29)$ (Kechris)

I'm referring to "Classical Descriptive Set Theory" by Kechris. I found this question on the same theorem $(8.29)$ pp. $49$, but I need further explanations about both the proof and a consequence of ...
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$S_\infty$ is a non-locally compact Polish group (Kechris)

Here is example $7)$, pp. $59$ of Kechris' book "Classical Descriptive Set Theory": Let $S_\infty$ be the group of permutations of $\mathbb{N}$. With the relative topology as a subset of $\...
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Given a non-measurable set and a cardinal does there exist a subset of that cardinality which is measurable?

Let $A\subseteq\mathbb{R}$ not be Lebesgue-measurable. My questions then are: If $\kappa\leq 2^{\aleph_0}$ is any cardinal such that $\kappa>\aleph_0$ does there then exist a measurable subset $M$ ...
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Meagerness on an open subset with the relative topology (Kechris)

I'm still working on meagerness, and what I post is something that I had achieved yesterday but today I don't remember how I did that. At pp. $48$, Kechris (Classical Descriptive Set Theory) defines, ...
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Baire Property on Baire Spaces (Kechris' book)

Kechris states the following result in "Classical Descriptive Set Theory", pp. $48$: ($\boldsymbol{8.26}$) Proposition Let $X$ be a topological space and suppose that $A\subseteq X$ has the BP (...
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Is a weak basis of a topological space a basis and vice versa? (Kechris' book)

I'm reading Kechris' book "Classical Descriptive Set Theory" and the author gives the following definition (pp. $49$, row $3$): A weak basis of a topological space $X$ is a collection of nonempty ...
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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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 $...
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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 ...