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 every set comeagre in its closure?

Let $A$ be a set in a topological space $X$. We know that $A$ is dense in its closure $\bar{A}$. This implies that $\bar{A} \setminus A$ is nowhere dense in $\bar{A}$ (using a characterisation of ...
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29 views

When Borel functions and Baire functions are equal?

Suppose $X$ is compact metric space. Let $A$ be the smallest set of complex functions containing all continuous functions such that: If $f_n \in A$ are uniformly bounded and $f_n \to f$ pointwise ...
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69 views

Proof of Kondô-Addison theorem

The proof of the (lightface) Kondô-Addison theorem (aka $\Pi^1_1$ uniformization) that I know goes like this: for a $\Pi^1_1$ set $R \subseteq 2^\omega \times 2^\omega$, define the uniformization of ...
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132 views

How does the cardinality of the set of all probability measure on a set $X$ change according to the cardinality of $X$?

I was wondering concerning the following problem: Take $X$ as a parameter space endowed with its Borel $\sigma$-algebra. What is the cardinality of $\Delta (X)$, understood as the set of all ...
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75 views

an application of Martin's axiom to Lebesgue measure

I am an beginner in set theory and try to solve an exercise in the 2nd chapter in Kunen's book: Assume $MA(\kappa)$, let $A$ be a family of Lebesgue measurable subsets of $\mathbb{R}$,with ...
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50 views

Constructing a Borel-measurable function from a functional inequality

Let $(X, \mathcal{X})$ and $(A, \mathcal{A})$ be standard Borel spaces, and let $q(\cdot | \cdot)$ be a substochastic kernel with source $(X\times A, \mathcal{X}\otimes\mathcal{A})$ and target $(X, ...
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41 views

Baire property and perfect set

Be $A\subset X$ whit the Baire property and not meager. Show that $A$ contain a subset perfect nonempty. I try prove that $A$ contain a subset $G_{\delta}$ no-numerable and use the theorem of Cantor ...
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44 views

Constructing a Borel-measurable function from a semi-analytic one

Consider a function $f: X \rightarrow (0, \infty)$ whose domain $X$ is a standard Borel space. Suppose $f$ is upper semi-analytic, i.e. for every $\lambda \geq 0$ the set $\{x \in X : f(x) > ...
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43 views

Prove the function is a homeomorphism.

Let $X$ be a Polish space with a complete compatible metric $d$. Let $G$ be the group of all isometries of $<X,d>$ with pointwise convergence topology and the composition operation. Fix a ...
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1answer
47 views

Omega-model of WWKL consisting of random reals

I've been trying to show, as an exercise, that over $\mathrm{RCA_0}$ weak weak Kőnig's lemma (WWKL) does not imply weak Kőnig' lemma (WKL). I've been working on it by constructing an $\omega$-model ...
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169 views

Can a basis for $\mathbb{R}$ be Borel?

Work in ZF (so no choice). Then it is consistent that there is no (Hamel) basis for $\mathbb{R}$ as a $\mathbb{Q}$-vector space. My question is about models where $\mathbb{R}$ does have a basis, but ...
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195 views

Spanning the reals with a small set - choicelessly

Working in ZF (so, no choice): is it possible that there is a set of reals $X$ such that $\vert X\vert<\mathbb{R}$, but $X$ generates $\mathbb{R}$ as a subgroup under addition? This seems ...
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37 views

Determinacy and uniformization

It's known that $PD$ implies projective uniformization. Assuming $AD$, is there an analogous theorem that holds for all subsets of the plane (where the uniformizing functions are reasonably definable ...
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1answer
43 views

Question about comeager set in a Polish space

I'm trying to understand a proposition of A. Kechris in chapter 8 of his Classical Descriptive Set Theory, in which given a non empty metrizable separable space $X$ that is dense in a Polish space ...
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1answer
32 views

Equivalence between two comeager sets related to free groups

Let $G$ be a non discrete Polish group. For every $n\ge 2$ equip $G^n$ with the product topology. Saying that $F_n=\{(g_1,\dots,g_n)\in G^n: \{g_1,...,g_n\}$ freely generates a free subgroup of rank ...
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70 views

a topological property of the product topology

Let $G$ be a non discrete Polish group. Let $K$ be a compact set of $G$, $C$ a closed set of $G^n$ and $B$ an open set of $G^n$. Suppose $K^n\cap C\subseteq B$. Prove that there is an open set of $G$, ...
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125 views

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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43 views

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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60 views

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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49 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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46 views

The intersection of comeager sets in a Baire Space [closed]

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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52 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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117 views

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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138 views

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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62 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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90 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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85 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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14 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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24 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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23 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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58 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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37 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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1answer
35 views

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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48 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
56 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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41 views

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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76 views

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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222 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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1answer
54 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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20 views

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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80 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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68 views

$\mathbb{R}^3$ \ $\mathbb{Q}^3$ 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
173 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
68 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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30 views

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 ...