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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Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary ...
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$G_\delta$ subgroups of a Polish Group

Let $X$ be a Polish Group. It's known that every its Polish subgroup is a $G_\delta$. Pick one of them, say $V$. Is it true that $V$ is the intersection of open subgroups? Thank you
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Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
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Can a Banach measure be consistent with Friedman's Fubini-type theorem for non-measurable functions?

I read on Wikipedia that Harvey Friedman proved that the following is consistent with ZFC+¬CH: For all functions $f:[0,1]^2 \mapsto \mathbb{R}^+$ such that both $\int_0^1 \left ( \int_0^1 f(x,y) dy ...
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34 views

Every comeager set in a perfect Polish space contains an uncountable dense $G_\delta$ set

Let $X$ be a perfect Polish Space. Prove that every comeager contains an uncountable dense $G_\delta$ set. It's known that every perfect Polish Space has cardinality $2^{\aleph_0}$. It's easy to ...
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23 views

on comeager sets of a Baire Space

Is it true that in a Baire Space the intersection of two comeager sets is not empty? If yes, is the intersection comeager too?
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open dense subset of a Baire Space

Let $X$ be a Baire Space and let $Y$ be a comeager subset of $X$. Is true that $Y$ contains an open dense subset of $X$? Thank you
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20 views

on the hyperspace K(X) with Vietoris topology

Let $X$ be a Polish Space without isolated points. Let $K(X)$ be the sets of the compact non empty subsets of $X$ equipped with the Vietoris topology. Can we say that also $K(X)$ has no isolated ...
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46 views

uncountable co-meagre set in Polish Spaces

Let $X$ be an uncountable Polish Space and let $Y$ be a co-meagre subset of $X$. How can I prove that $Y$ is uncountable? Possibly proof without using borel sets. Thank you
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51 views

Structure of the $L_1$ space of measurable subsets of $[0,1]$

Let $\mathcal A$ be a Borel $\sigma$-algebra on $[0, 1]$, and let's introduce a metric on it by $$ d(A, B) = \lambda(A\mathbin\Delta B) \qquad \forall A,B\in \mathcal A $$ where $\lambda$ is the ...
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What can the reals of an inner model be?

This is probably a silly question. Call a set of reals $X$ a constructibility ideal (in analogy with a Turing ideal) if $X$ is closed under effective join $r\oplus s: n\mapsto 2^{r(n)}3^{s(n)}$ and ...
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Good text to start studying topological games?

Topological games and some similar infinite games seem to be often used used as a tool in some areas of general topology, but also some other areas, such as Ramsey theory, filters, etc. Probably the ...
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52 views

Measurable projection theorem proof reference

I'm beginning to study about stochastic processes, and currently focusing on stopping times and hitting times. The textbook I'm using is "Stochastic Integration Theory" by Medvegyev (and Karatzas ...
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1answer
86 views

There is no Baire bijection between $\mathbb R$ and the set of functions $\mathbb Z\to\mathbb R$ modulo shifts

Let $X$ denote the set $\mathbb{R}^\mathbb{Z}$ (the set of all functions from integers to reals), and $\sim$ the equivalence relation on $X$ defined by: $f \sim g$ iff there is a $z \in \mathbb{Z}$ ...
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82 views

Can you define a sensible probability measure on the set of countable transitive models of ZFC?

It is well known that the set of hereditarily countable sets $H(\omega_1)$ —or, if you prefer, $H_{\omega_1}$— has cardinality $2^{\aleph_0}$, and I understand that every countable ...
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12 views

Use the $G_0$ dichotomy to prove the Lusin separation Theorem

Use the $G_0$ dichotomy to prove the Lusin separation theorem. (Note though that the Lusin separation theorem is used repeatedly in the proof of the G0 dichotomy.) Hint. If $A, B ⊂ X$ are disjoint ...
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Let $G$ be an open graph on a Polish space $X$.

Let G be an open graph on a Polish space $X$. Show that either $X$ is a union of countably many closed $G$-anticliques, or there is a perfect $G$-clique.
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Paramerized $G_0$ dichotomy

Suppose that $X, Y$ are Polish spaces and $G$ is an analytic graph on $X × Y$. Show that either $X × Y$ can be covered by countably many Borel sets $B_n ⊂ X ×Y$ for $n ∈ ω$ such that each vertical ...
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22 views

Show there is a continuous function.

Let $B ⊂ ω^ω × ω^ω$ be a Borel set. Show that there is a total continuous function $f : ω^ω × ω^ω$ such that either $f ⊂ B$ or $f ⊂ (ω^ω × ω^ω \setminus B)^{−1}$. Is this statement true with the ...
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22 views

Nowhere integrable function [duplicate]

Does there exist a function $f : I \to \mathbb{R}$ defined on an interval $I \subseteq \mathbb{R}$ that is measurable but not integrable on any compact subinterval $[a,b] \subseteq I$? One can try to ...
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46 views

Maximum Baire class of a Riemann integrable function

In this answer (see example 1), Andrés Caicedo gives an example of a function $f : [0,1] \to \mathbb{R}$ which is Riemann integrable and is (strictly) of second Baire class. Are there Riemann ...
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35 views

Relation between the Effros structure and Vietoris topology

Let $(X,\tau)$ be a topological space (eventually, a polish space) and $\mathcal{F}$ the collection of all closed sets of $X$. Given $\mathcal{U}\subseteq \tau$ finite, define $$ ...
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A Ramsey not Completely Ramsey Set of $[\omega]^\omega$.

Let $a \in [\omega]^{<\omega}$ (a finite subset), $A \in [\omega]^{\omega}$ (an infinite subset). Let us define $$[a, A] = \{a \cup B: B \in [A]^{\omega} \wedge max(a) < min(B) \}.$$ These ...
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139 views

“Clubbiness” of $\Pi^1_n$ sets

I'm sure this is just my google-fu failing me, but: what are sufficient large cardinal axioms to guarantee "Every (boldface) $\Pi^1_n$ set of countable ordinals contains or is disjoint from a club ...
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84 views

Prove there is a Bernstein set $B$ such that $B+B$ is also Bernstein

Show that there exists a Bernstein set $B$ such that $B+B$ is also Bernstein. I have tried to use the definition that neither $B$ nor its complement contain a perfect set.
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66 views

Is there a borel set $A$ and a linear map $f$ such that $f(A)$ is not borel?

Is there a borel set $A\in\mathbb{R}^n$ and a linear map $f:\mathbb{R}^n\to\mathbb{R}^n$ that $f(A)$ is not borel set? I think there is but I can't find it.
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45 views

Show that there exist Borel sets $B_n$ such that $B=\bigcup B_n$

Let $X$ be a Polish space. Let $B$ be a Borel subset of $X \times X$ with the following property: $$\forall x \in X \ \left|\left\{y : (x,y) \in B\right\}\right| = \aleph_0.$$ Show that there exists ...
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1answer
52 views

Measure zero on all Fat Cantor Sets

Let $F_n\subset [0,1]$ be a Fat Cantor Set (so that $[0,1]\setminus F_n$ is dense) of Lebesgue measure $1 - 1/n$, and let $F = \bigcup_n F_n$. Does there exist a probability measure $\mu$ on $[0,1]$ ...
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Finite union of Polish spaces is not Polish?

Schwartz writes in his book "Radon Measures on Arbitrary Topological Spaces", p. 110: "But even a finite union of polish spaces need not be polish". The same statement can be also found here. How ...
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Not a small, not a big set

Consider a unit interval $X = [0, 1]$ endowed with Borel $\sigma$-algebra generated by the usual topology. I am looking for a measurable set $K\subseteq X$ which has a $0$ Lebesgue measure, but such ...
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217 views

Existence of a section of non-zero measure

Let $X$ and $Y$ be measurable spaces, and $A \subseteq X\times Y$ is a measurable subset of the product space. For any $y\in Y$ let $A_y = \{x\in X: (x,y)\in A\}$ be the $y$-section of $A$. Under ...
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48 views

Classification of Polish topologies on a countably infinite set

Let $X$ be a countably infinite set. While investigasting the literature on Polish spaces, I met so far only examples for compact or locally compact Polish topologies on $X$: the order topology on ...
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For a Borel action of a locally compact second countable group G on a standard Borel space S, are the orbits always Borel?

In his book "Ergodic Theory and Semisimple Groups" Robert Zimmer opens Chapter 2 by discussing the situation of a locally compact second countable group $G$ with a Borel action on a standard Borel ...
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75 views

$\Pi^1_1$ singletons and $\Delta^1_2$ wellorders on $\omega$ in $L$

I have been trying to show the supremum $\delta^1_2$ of ordinals that are $\Delta^1_2$ wellorders on $\omega$ is exactly equal to the least ordinal $\delta$ such that $L_\delta$ contain all $\Pi^1_1$ ...
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$\mathbb{R}^3$ \ $\mathbb{Q}^3$ is a union of disjoint lines, The lines are not in an axis direction. [duplicate]

I have a folowing question: $\mathbb{R}^3$ \ $\mathbb{Q}^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski. Any can give ...
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1answer
164 views

$\mathbb{R}^3$ \ $\mathbb{Q}^3$ is union of disjoint lines. The lines are not in an axis diretion.

I have the following question: $\mathbb R^3\setminus \mathbb Q^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski. ...
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2answers
61 views

Do (the restrictions of) continuous maps (to the rationals) form a Borel set?

Consider, within the Polish space $\mathbb{R}^\mathbb{Q}$ (with product topology), the subset of all those maps that can be extended to a continuous map on all of $\mathbb{R}$. It's easy to see that ...
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Why is the Dynkin system generated by open intervals in $\mathbb{R}$ the Borel $\sigma$-algebra?

Suppose $\mathcal{B}$ is a collection of subsets of $\mathbb{R}$ which contains the open sets, and is closed under complements and countable disjoint unions. Then $\mathcal{B}$ contains the Dynkin ...
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54 views

Is a bijective projection function measure preserving?

A subspace with dimension strictly less than the dimension of vector space has (Lebesgue) $measure=0$. Let $V$ be a vector space with $dimension=n$. To show that some set $S$ in V is zero-measure, is ...
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35 views

Vanishing Cantor-Bendixson derivative if and only if scattered

In the question asked here, it is claimed in a comment and answer of Brian M. Scott that a subset of $\mathbb{R}$ is scattered if and only if it has vanishing Cantor-Bendixson derivative. I thought ...
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Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...
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Why is the intersection of countably many homogeneously Suslin subsets of $\,^{\omega} \omega$ homogeneously Suslin?

I found this assertion in these notes: The derived model theorem (Steel) right in the beginning on page 3, together with the remark that this is 'not too hard to show'. Unfortunately, I'm ...
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Are Lusin and Souslin spaces sequential or even Fréchet-Urysohn?

First some definitions: A Polish space is a separable and completely metrizable topological space. A Hausdorff space is Lusin if it is the image of a Polish space under a bijective continuous map. A ...
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Members of $\Sigma^1_4$ sets

I'm pretty sure this is an easy descriptive set theory question that I'm just blanking on. Is it consistent with large cardinals - say, with a measurable - that every (nonempty) $\Sigma^1_4$ class ...
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Modifying definition of $\sigma$-algebra

Let's start with basic definitions. Def. Given a set $X$ we say that $\Sigma\subset 2^X$ is a $\sigma$-algebra, if $X\in\Sigma,$ $\Sigma$ is closed under complementation and countable unions. ...
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38 views

Why the set $\{f \in \omega^\omega : \exists_{m}\forall_{m \leq n}^{}(f(n)\neq g(n))\}$ is $F_{\sigma}$ and meager?

For any $g \in \omega^\omega$, Why the set $\{f \in \omega^\omega : \exists_{m}\forall_{m \leq n}^{}(f(n)\neq g(n))\}$ is $F_{\sigma}$ and meager?. I do not know why this set is $F_{\sigma}$ and ...
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61 views

Intersection of Borel subset with line

Let $y \in \mathbb{R}$. If $A$ is a Borel subset of $\mathbb{R}^2$, then $$A(y) = \{x \in \mathbb{R}| (x,y) \in A\}$$ is a Borel subset of $\mathbb{R}$. I think that I have to show that $A(y)$ is in ...
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57 views

Self-Dual Pointclass with a Universal Set?

I'm a little bit embarrassed to be asking this, but I can't find my mistake! Assume AD. I am going to construct a self-dual pointclass with a universal set, swhich is impossible. Let A $\equiv_w A^c$ ...
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46 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
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396 views

Definition of infinite tree in set theory

Really basic question concerning trees in set theory. What is the definition of an infinite tree? I ask the following because, rather peculiarly, neither in Kechris classical book on descriptive ...