# Tagged Questions

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