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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Does this system of open sets have to cover the whole space?

I have been studying basics of descriptive set theory lately. In the lecture notes I follow (sadly, the notes are written in Czech), there is the following definition: Let X be a topological space....
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14 views

How to prove that these sets are $\mathbf{\Delta}_\alpha^0$.

I have been studying basics of descriptive set theory lately. In the lecture notes I follow (sadly, the notes are written in Czech), there is a lemma (which is used to prove that the functions of ...
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1answer
19 views

Why for every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets?

$f$ is a real-valued baire class $\xi$ function if $\{ f < c \}$ and $\{ f >c \}$ are in $\sum_{\xi +1}^0$ for every $c \in \mathbb{R}$. In the proof of Proposition $5.2$ page $24$, we have ...
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Convert Baire class $\xi$ function into Baire class one

Let $f$ be a real-valued baire class $\xi$ function. In this paper, page $24$, section $5$, before remark $5.1$, the author defined the set $T_{f,\xi}=\{ \tau^{\prime} : \tau^{\prime} \supseteq \...
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51 views

Consequences of the principle of dependent choices (DC)

It is known that if we assume the axiom of determinacy every set of real numbers is lebesgue measurable. In order to study this, I'm following Jech's Set Theory book. There, Jech says that apart from ...
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47 views

Amalgam of trees

Definition A tree is a partially ordered set $(T, <)$ such that for each $t \in T$, the set $\{s \in T : s < t\}$ is well-ordered by the relation $<$. For trees $(T,<_T)$, $(S,<_S)$, $...
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101 views

Small filters are measurable

i want to show, that a filter $\mathcal{F}$ on $\omega$ (considered as a subset of $2^\omega$), which is small, is measurable. I found a lemma (without proof), that every small set is null. So, if $\...
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57 views

Borel set as union of $G_\delta$ and countable set

What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?
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20 views

Are the basic open sets of the Baire space closed?

One way to describe the topology of the Baire space $\mathbb{B} = \omega^\omega$ is that the basic open sets are of the form $N_\eta = \left\{ f \in \omega^\omega \middle |\ \eta \subseteq f \right\}$...
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45 views

Any $G \subseteq \mathbb{N}^\mathbb{N}$ is clopen

I have a question concerning the space $\mathbb{N}^\mathbb{N}$. I found in Srivastava's book on Borel sets that the sets of the form $$\Sigma (s) := \{ \alpha \in \mathbb{N}^\mathbb{N} \ | \ s \prec ...
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Is a non-meagre set comeagre in its closure?

Let $A$ be a non-meagre subset of a topological space $X$. Is $A$ comeagre in its closure $\mathcal{Cl}(A)$?
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44 views

The set of fixed points of a Borel function

Let $f: \mathbb R \rightarrow \mathbb R$ be a Borel function, is the set $\{x : f(x)=x\}$ Borel? Edit: As short questions seem to be quite unpopular here, I'll elaborate a little: As the graph of the ...
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18 views

Existence of a polish group $G$ such that $G\cong\text{Isom}(G)$.

It is known that every polish group is isomorphic to $\text{Isom}(X)$ for some polish space $(X,\tau)$. My question is if there exist some infinite polish group such that $G\cong\text{Isom}(G)$. ...
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85 views

Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...
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Proof of Borel-Wadge determinacy without using Borel determinacy?

It's easy to prove Borel-Wadge determinacy from Borel determinacy. But it's often said that Borel-Wadge determinacy is 'much weaker' than the latter. This is then argued by showing models in which the ...
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1answer
55 views

What is a Dynkin system? ($\lambda$-system)

Until recently, all my knowledge of measure theory and Lebesgue integration are from Rudin's book, which focuses solely on the Lebesgue measure, its construction and nothing else. I have just put my ...
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60 views

Proof that the Cardinality of Borel Sets on $\mathbb R$ is $c$ without using the ordinals .

I'm trying to prove that cardinality of Borel sets is $c$ without using the concept of Ordinal number ! I know that the Cardinal of Borel sets are greater than $c$ because of every point in $\mathbb R$...
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38 views

If $A \subset \mathcal{N}^2$ is a $\mathbf{\Sigma}^0_\alpha$ set, then $\{x : (x,x) \in A\}$ is also $\mathbf{\Sigma}^0_\alpha$.

This is the boldface Borel hierarchy on Baire space. Jech states this with a "clearly". What am I missing that makes the statement completely obvious? I clearly have zero intuition for this material....
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21 views

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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35 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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77 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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137 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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93 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 $|A|=\...
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1answer
51 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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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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50 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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1answer
44 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
52 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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261 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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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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41 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 $Y$...
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1answer
33 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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1answer
71 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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1answer
152 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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47 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 $S_\...
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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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1answer
51 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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49 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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53 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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1answer
142 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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65 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
95 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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89 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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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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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 ...