Questions tagged [descriptive-set-theory]
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 there a function $f :A^{<\mathbb{N}}\to E$ such that $f (s{}^\frown a)=h(f(s),a,|s|)$ for any $s\in A^{<\mathbb{N}}$ and $a\in A$?
Given a set $A$, define $A^{<\mathbb{N}}:=\cup _{n\in\mathbb{N}}A^n$ with $A^0:=\{\emptyset\} $ and $A^n$ being the $n$-fold cartesian product of $A$.
For any $s:=(s_0,\cdots,s_{n-1})\in A^n\...
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Formalizing the construction of a Cantor scheme
I'm asking this question here since a similar question asked on math.stackexchange didn't receive an answer.
I'm trying to understand the proof of the following theorem:
Theorem (Brouwer): The Cantor ...
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What's the intuition behind the definition of "rank of a tree"?
Let $T$ be a well-founded tree on $\mathbb{N}$, that is, the set of infinite branches of $T$ is empty. Define a function $\rho_T : T\to \mathbf{ON}$ inductively by $$\rho_T(u)=\sup\{\rho_T(v)+1: u\...
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Complement of any dense countable subset of reals is homeomorphic to irrationals
I recently stumbled upon this:
For any infinite countable subset $A\subseteq\mathbb R$ such that $\overline A=\mathbb R$, the complement $\mathbb R\setminus A$ is homeomorphic to the Baire space. (Or,...
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A question on an equivalent form of Kalmar hierarchy
In the book named A Course on Borel Sets by Srivastava, the set of all clopen subsets of $\mathbb{N}^\mathbb{N}$ is given as in the following screenshot. However, the book doesn't provide with a ...
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The relation between the cardinality of Bore $\sigma$-algebra and axiom of choice
Under axiom of choice, the cardinality of Borel $\sigma$-algebra $B$ is $\mathfrak{c}$.
In this proof axiom of choice is used three times: To prove
$\omega_1$-times recursion is sufficient,
each $|B_{...
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Help in understanding this example in Kechris on the Borel heirarchy
I am trying to read Kechris's descriptive set theory (self-study only). In Chapter 22 on the Borel heirarchy, let $A \subseteq \mathbb{Q}$ be such that $A$ and $\mathbb{Q} \setminus A$ are dense (I ...
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A Borel set with convex sections has a Borel projection
The following is an exercise from "A Course on Borel Sets" by S.M. Srivastava.
Exercise 4.7.10 Let $X$ be a Polish space and $B \subseteq X \times \mathbb R^n$ a Borel set with convex ...
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Invertibility of a measurable mapping from lower and upperbounds on the induced pushforward measure
Let $\Omega \subseteq \mathbb{R}$ be open and consider the standard Borel space $(\Omega, \mathcal{B}(\Omega), \mu)$, where $\mu$ denotes the Lebeasgue measure. Let $f: \Omega \to \Omega$ be a ...
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Is the set of all antichains of $\omega^{<\omega}$ avoiding all chains Borel?
Let's identify $\mathcal P(\mathbb N^{<\mathbb N})$ with $2^{\mathbb N^{<\mathbb N}}$, so it is a Polish space. The set $\mathcal A=\{A\subseteq \omega^{<\omega}: A \text{ is a maximal ...
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Is there a sequence of two-set partitions of $[0,1]$ such that the partition generated by any subsequence is a partition into finite sets?
More specifically, I'm wondering if there is a sequence of partitions $\{\mathcal{P_n}\}_{n\in \mathbb{N}}$ of the unit interval $[0, 1]$, where each $\mathcal{P}_n$ consists of precisely two sets, ...
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Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
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Clarification on "absolute" property
I read that if some property $P(x)$ defined by a formula $\phi$ is absolute for some class $M$, then $\phi(P(x)) \leftrightarrow \phi^M(P(x))$, where $\phi(P(x))$ is interpreted in $V$, and $\phi^M(P(...
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Equivalence of definitions of "standard Borel space"
I met the following definition of standard Borel spaces in Durrett's probability theory book (slightly rephrased):
$(S,\mathcal{S})$ is said to be standard Borel if it is isomorphic (as a measurable ...
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Help in understanding this part of Mansfield / Weitkamp
I am trying to read Effective Descriptive Set Theory by Mansfield and Weitkamp (self-study only so please bear with me ... am a beginner in this area).
I need help in understanding Example 1.24 of the ...
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