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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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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3answers
326 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 ...
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1answer
36 views

space of all lipschitz maps is a polish metric space

Suppose that $(X, d_X)$ and $(Y, d_Y )$ are Polish metric spaces. Let $L(X, Y )$ denote the set of all Lipschitz maps from $X$ to $Y$ with the pointwise convergence topology. Show that $L(X, Y )$ is ...
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2answers
61 views

Is a particular countable subset of the Cantor set Polish?

Consider the Cantor space $\mathcal{C} := \{ 0, 1 \}^{\mathbb{N}}$ and the subset $\mathcal{T} \subseteq \mathcal{C}$ of sequences that start with $1$ and eventually "terminate" with $0$, i.e. ...
2
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0answers
62 views

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. ...
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15 views

Are some some particular subspaces of cadlag functions Polish?

Consider the space $D := D((0, \infty), \mathbb{N})$ of cadlag functions $f : (0, \infty) \to \mathbb{N}$ equipped with the Skorokhod $M_1$-topology. Then $D$ is Polish. Question 1: I want to check ...
2
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1answer
24 views

select a nested sequence from a $G_{\delta}$ set

Here, page $457$, in the proof of theorem $1$, there is this sentence 'Select a nested sequence $\{ U(n,i,j)| j=1,2,... \}$, of open subsets of $I$ whose intersection is $f^{-1}[\frac{i}{n}, ...
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0answers
43 views

Is there a name for functions $f$ such that $\{f \le a\}$ is analytic?

Let $X$ be a Polish space and $f : X \to \mathbb{R}$ an arbitrary function. In The limit inferior of Borel functions I showed that a certain function $f$ has the following property: For each $a ...
4
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1answer
47 views

The limit inferior of Borel functions [closed]

Suppose $X$ is a separable metric space and $F \colon X \times ℝ_+→[0,1]$ is Borel. Let $f(x) = \liminf_{ε→0} F(x,ε)$. Is $f$ Borel?
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15 views

$\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, $\mu_{\mathcal{C}}-$a.e

If $A \subseteq \mathcal{C}$ is $\mu_{\mathcal{C}}$-measurable, then $\lim_{n \to\infty}\frac{\mu_{\mathcal{C}}(A\cap N_{x|n})}{\mu_{\mathcal{C}}(N_{x|n})}=\mathcal{X}_{A}(x)$, ...
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27 views

Can we find a unique probability Borel measurable $\mu$ on $\mathcal{C}$ with $\mu(N_s)=\phi(s)$

If $\phi:2^{<\mathbb{N}}\to [0,1]$ satisfies $\phi(\emptyset)=1$ and $\phi(s)=\phi(s^{\widehat{}}0)+\phi(s^{\widehat{}}1)$ for all $s \in 2^{<\mathbb{N}}$. Can we find a unique probability ...
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2answers
57 views

Cantor-Bendixson rank of a closed countable subset of the reals, and scattered sets

I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the ...
1
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1answer
28 views

Product of Borel and non-Borel set

It is true that product of Borel and non-Borel sets is non-Borel set? More precisely, I would like to know if $V $×$ \{1\}$ is Borel, where $V$ is Vitali set.
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289 views

Continuous surjections onto $\mathbb{R}$

I have two questions about continuous functions: Suppose $X \subseteq \mathbb{R}$ and $X$ has same cardinality as $\mathbb{R}$. Can we find a continuous function from $X$ onto $\mathbb{R}$? Suppose ...
6
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2answers
157 views

A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
1
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1answer
53 views

Continuous function on Polish space

I want to prove the following and have no idea how to proceed: For a continuous function $f: X \mapsto Y$ where $X$ is Polish and $Y$ is Hausdorff the following are equivalent: $f[X]$ is ...
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1answer
42 views

Is the Borel-sigma-Algebra of a Polish space always countably generated?

Wikipedia says: Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. (Polish space, Wikipedia) So since the ...
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1answer
38 views

Is the Borel reduction of an hyperfinite equivalence relation still hyperfinite?

Let $(X,E)$, $(Y,F)$ be Borel equivalence relations. Assume $(X,E)\leq_{\text{B}}(Y,F)$.${}^{\dagger}$ Does it hold that: $F$ hyperfinite $\Longrightarrow$ $E$ hyperfinite?${}^{\ddagger}$ I do not see ...
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0answers
47 views

Undetermined game of length $\omega_1+\omega$, without choice

On the following page, Taranovsky is talking about his "Determinacy Maximum" axiom: http://web.mit.edu/dmytro/www/DeterminacyMaximum.htm He also justifies the choice of the name, by pointing out that ...
4
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1answer
62 views

Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
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182 views

The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need ...
2
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1answer
45 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
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1answer
26 views

definition of countably generated Borel space

I have one source http://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0089999-2/S0002-9947-1957-0089999-2.pdf page 137, first paragraph of section 2 which says a countably generated Borel ...
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2answers
87 views

Dense-in-itself open sets in a subspace of the real line

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of ...
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0answers
37 views

$f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of measurable functions

Let $X$ be a measurable space and $Y$ a Polish space. I want to show the following equivalence. $f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of ...
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1answer
42 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
2
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1answer
55 views

Lusin space, isolated point

I have a question about Lusin space. Definition A Hausdorff topological space, $(X,\tau)$ is said to be a Lusin space if, there exists a topology $\tau'$ (on $X$) stronger than $\tau$ such that ...
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1answer
63 views

Informal interpretation of meager sets

I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets. The general setup to tease out ...
2
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1answer
74 views

Pointclass of $\text{dom}(F)$ where $F:\omega^\omega\rightarrow\omega^\omega$ is partial recursive.

The definition I am working with: A partial function $F:\omega^\omega\rightarrow\omega^\omega$ is said to be partial recursive iff the partial function $G:\omega^\omega\times\omega\rightarrow\omega$ ...
4
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2answers
112 views

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
3
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1answer
48 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
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1answer
28 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
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1answer
39 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
5
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1answer
95 views

No Borel well-order of the reals?

I'm told there is no Borel well-order of the reals (in ZFC). I'm told, in fact, that this is because of Borel determinacy. However, this is usually a vague handwave of the form (a) take the usual ...
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1answer
25 views

$\{K∈K(X):K⊆U\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$

Let $X$ be a Polish space. The family of set $(i)$ $\{K∈K(X):K⊆U\}$ $(ii)$ $\{K∈K(X):K∩U≠∅\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$ where $K(X)$ is the space of all compact subsets of ...
2
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2answers
35 views

What's the difference between $ω^{<ω}$ and $ω^{ω}$?

I think that the first is all finite sequences of natural numbers, while the second is all infinite sequences of natural numbers. However I'm not sure how to draw the tree for $ω^{<ω}$, nor can I ...
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0answers
42 views

Are clopen sets Borel?

I understand that open sets, closed sets, $F_\sigma$-sets (countable unions of closed sets), $G_\delta$-sets (countable intersections of open sets), $F_{\sigma\delta}$-sets, $G_{\delta\sigma}$-sets ...
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1answer
28 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
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1answer
48 views

Borel image of a Polish space

Let $X, Y$ be Polish spaces and $f : Y \to X$ be a Borel function, i.e. preimage of every Borel subset of $X$ be a Borel subset of $Y$. Prove that $f[Y]$ is an analytic subset of $X$. Note: If ...
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1answer
21 views

Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. Where $\mathcal{C}=2^\mathbb{N}$.

If $X$ be a nonempty perfect Polish space, $Y$ a second countable space, and $f.X\to Y$ be inyective and Baire measurable. Question: Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. ...
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51 views

$A=\bigcup_{i \in I}A_i$ is meager.

Let $X$ be a Polish space. Let $(I,<)$ be a wellordered set and $(A_i)_{i \in I}$ a family of sets meager in $X$. Let $A=\bigcup_{i \in I}A_i$. If consider the relation $x \leq^{*} y$ defined by: ...
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0answers
46 views

Show that there is a Borel function from [0,1] to $2^ω$

Show that there is a Borel function from [0,1] to $2^ω$. Conclude that there is a Borel function from $[0,1]^ω$ to $2^ω$. I'm not sure how to go about this. I know that a function is Borel if the ...
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0answers
48 views

Every perfect Polish space is a continuous 1-1 image of the Baire space

I am learning descriptive set theory by myself and stumble upon this theorem without proof, could somebody help me prove it: Every perfect Polish space is a continuous 1-1 image of the Baire space ...
3
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1answer
64 views

A $G_δ$ subset of $2^ω$ that is homeomorphic to $ω^ω$

How do I show that there is a $G_δ$ subset of the Cantor space $2^ω$ that is homeomorphic to the Baire space $ω^ω$? I've been given the hint to consider $G = \{x ∈ 2^ω : x\text{ is not eventually ...
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0answers
28 views

How many Borel sets are on an infinite Polish Space?

I am learning Descriptive set theory, and my question is how many Borel sets are on an infinite Polish space? I think they have the cardinality of the continuum, but I just do not see how to prove ...
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23 views

Understanding C-universal classes

There is a theorem in the book Classical Descriptive Set Theory, A. Kechris. In page 168 he states the following theorem: Let $X$ be a separable metrizable space. Then for each $\xi \geq 1$, there is ...
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29 views

$\text{BP}(X)/\text{MGR}(X)$ be the quotient space $\{[A]:A \in \text{BP}(X)\}$

Let $X$ be a topological space. $(a)$ $\text{BP}(X)$ denote σ-algebra of subsets of $X$ with the Baire Property $\text{BP}$. $(b)$ $\text{MGR}(X)$ denote the σ-ideal of meager sets in $X$. Let ...
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1answer
57 views

The inverse image of dense set is dense and of a comeager set is comeager?.

Let $X,Y$ be topological and $f:X \to Y$ be open and continuos. I am studying Baire space and I would like to try the following facts : $(i)$ The inverse image of dense set is dense. ...
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2answers
32 views

$X\times Y$ is second countable Baire spaces

If $X,Y$ are second countable Baire spaces, so is $X\times Y$. ? If true, a suggestion to try thanks
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32 views

measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...