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 the following set Borel? [closed]
Define the set
$$
A = \{ x \in \mathbb{R} \, \vert \, \forall M \in \mathbb{R}: \exists p, q \in \mathbb{Z}: p> M \text{ and }0 < \vert x-\frac{q}{p} \vert < \frac{1}{p^{3}} \}.
$$
Is this ...
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Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.
If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
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Show that the following metric is complete on $L_1(X,Y)$
I am working through the exercises of Kechris' Classical Descriptive Set Theory for personal edification. I've solved Exercise 3.5, but the proof was two pages. Did I miss a more straightforward ...
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Are projective sets from descriptive set theory projective objects in Set category?
There are projective sets in descriptive set theory. For them, the axiom of determinacy is not contradicting the axiom of choice.
Given the axiom of choice, every set is a projective object. But in a ...
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Why isn't $[0,1]^{\aleph_1}$ isomorphic to $[0,1]$ as a measure space.
I was talking with a professor, and he mentioned that $[0,1]$ (with Lebesgue $\sigma$-algebra and Lebesgue measure) isn't isomorphic as a measure space to $[0,1]^{\aleph_1}$ (with the product measure)....
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Alpha recursion - Constructible universe and Analytical hierarchy
Alpha recursion and Constructible universe are very intertwined, because the first is based on the concept of admissible ordinal $\alpha$ which is defined as an ordinal such that $L_\alpha$ - a set ...
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Two set-theoretical assumptions
My question about the following two set-theoretical assumptions:
union of less than continuum many meager subsets of $\Bbb R$
is meager in $\Bbb R.$
Union of less than continuum many meager subsets ...
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Given $p\in[0,1]$ form a set $\Omega$ and a function $f$ such that $\{f = 1\} = p$ and $\{f = 0\} = 1-p$
Given $p\in[0,1]$ is it always possible to form a set $\Omega$ and a function $f$ such that the expressions below are true?
$$
\begin{align}
\frac{|\{\omega\in\Omega\,:\, f(\omega) = 1\}|}{|\Omega|...
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Diagonal of the power of a standard Borel space is isomorphic to the original space
I don't have much experience with measure spaces, so I am asking for verification of some proofs. Thank you.
Let $(X, \Sigma)$ be a standard Borel space. Consider the product space $(Z, \Sigma') = (X, ...
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Dense co-dense subsets of a subspace of the Baire space
Let ${\mathbb{N}^\mathbb{N}}^*$ denote the infinite sequences in Baire space that do not have a periodic tail, which is a $G_\delta$ subset so it is a Polish space. Consider the equivalence relation ...
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Is the family of co-dense subsets of a space ideal?
A set $A$ in a space $(X,\tau)$ is called co-dense if $int(A)=\emptyset$. In this paper, it is mentioned the statement "ideal of co-dense". I am asking, how the family of co-dense subsets of ...
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Why does a zero'th order quantifier not increase the complexity of a $\bf{\Sigma}^1_1$ or $\bf{\Pi{^1_1}}$ assertion?
If $\mathcal{X}$ is a perfect product space, and $P\subset\mathcal{X}$ is a $\bf{\Sigma}^1_1$ or $\bf{\Pi}^1_1$ pointset, then why does adding a bounded existential quantifier keep it $\bf{\Sigma}^1_1$...
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Product with a discrete topology
Suppose $X$ is a countable set. Consider $(X, \tau_x)$ with the discrete topology. Denote another Polish topological space as $(Y, \tau_y)$, which is not neccesarily the discrete topology.
Consider a ...
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Is there a transfinite version of Post's Theorem?
Let $\emptyset^{(n)}$ denote the $n$th Turing jump of the empty set. Post's theorem states:
A set $B$ is $\Sigma^0_{n+1}$ if and only if $B$ is recursively enumerable by an oracle Turing machine with ...
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Why does the measure of the geometric difference yield a complete metric space on the $\sigma$-algebra?
I was talking with a professor, and he mentioned the following theorem:
Let $(X,\mathcal{F},\mu):=([0,1],\mathcal{B},\lambda)$ be the interval with the usual Lebesgue measure on the Borel sets. For $A,...
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Refine product topology to make Borel sets be clopen
I'm working on Exercise 2.28 in Prof. David Marker's notes http://homepages.math.uic.edu/~marker/math512/dst.pdf on refining the topology to make Borel sets clopen.
Question: Suppose $X$ is a Polish ...
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Which countable ordinals are common to every $\omega$-model of ZF?
Consider the progression of larger and larger countable ordinals ($\omega$, $\omega^{\omega}$, $\epsilon_0$, the Veblen hierarchy, etc., described nicely here).
On the one, hand, it seems clear that ...
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A erroneous Suslin scheme
Surely the question has an easy answer I'm not able to find, but anyway...
$R$ is the real line and $N$ the set of natural numbers, $I=N^N$.
Let $X\subset R$ be indexed by $I$. For every $a \in I$ be $...
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Is the rank of a tree a natural number or $\omega$?
Definition
Let $\mathcal{T}$ be the family of subsets $T$ of $S=\bigcup_{n\geq 1}\mathbb{N}^n$ such that $\sigma \mathord{\upharpoonright} k\in T$ whenever $\sigma\in T$ and $1\leq k\leq \#(\sigma)$. ...
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How does this look like a tree?
In a book there is a definition like that:
Let $\mathcal{T}$ be the family of subsets $T$ of $S=\bigcup_{n\geq 1}\mathbb{N}^n$ such that $\sigma \mathord{\upharpoonright} k\in T$ whenever $\sigma\in ...
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Why are finite stages of the Borel hierarchy $\bf{\Sigma}^0_n$ closed under bounded quantification?
Moschovakis proves in Theorem 1C.2 that the finite Borel pointclasses are closed under various operations, like continuous substitution, $\lor,\land,\neg$, etc. He runs the whole proof by induction, ...
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Are Borel sets whose projections are Borel closed under intersection?
The projection of a Borel set of $\mathbb{R}^n$ needn't be Borel, although the projection of a closed set is a countable union of compact sets, hence Borel.
My question is if $A,B \subset \mathbb{R}^n$...
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Shoenfield absoluteness lemma with just $\mathsf{ZF}$?
Shoenfield's absoluteless lemma asserts that if $\sigma$ is a $\Sigma_2^1$ statement (in arithmetic), then it is absolute to all inner models of $\mathsf{ZF} + \mathsf{DC}$. I am aware that this ...
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What condition(s) on $X$ and $A$ can ensure the existence of an element $a\in A$ such that $d(x_0, a)=d(x_0, A) $?
Let $(X, d) $ be metric space and $A\subset X$ and $x_0\in X$
Define $d(x_0, A) =\inf\{d(x_0, a) :a\in A\}$
What condition(s) on $X$ and $A$ can ensure the existence of an element $a\in A$ such that $...
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Is AC invoked in the claim "the product/sum of a sequence of separable metrizable spaces is separable"?
The Claim:The product/sum of a sequence of separable metrizable spaces is separable.
Here the sum of a family $((X_i, d_i))_{i\in I}$ of metric spaces is defined (up to isometry) as follows: By ...
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Is this subset of a closed set Borel measurable?
Let $V \subset [0,\infty)^n$ be closed. I've been studying the following subset of $V$ for research, which has useful properties, but cannot prove or disprove it is Borel measurable.
Introduce the ...
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What property does an identity relation has?
Reflective is obvious, but what about the other properties?
It's weird when we check the symmetric property of the identity relation. The symmetric property happens when (in a matrix representation) ...
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Prove that, in $\mathbb{R}$, $F_\sigma \neq G_\delta$, $F_{\sigma\delta} \neq G_{\delta\sigma}$, etc.
The author of the book I'm reading wants me to use the following lemma:
There exists an open set in $\mathbb{R}^2$ such that every open set in $\mathbb{R}$ is some vertical section in it, there exists ...
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Analytic sets have the Baire property
I am preparing for my Analysis qualifier. This problem is not from a past exam but from the text book Bruckner, A. W., et. al., Real Analysis (2nd edition), page 459.
A set in $A$ a topological space $...
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What exactly is the "$\mathbb{R}$" in $L(\mathbb{R})$?
In $L(\mathbb{R})$, presumably $L_0(\mathbb{R})$ is the transitive closure of "$\mathbb{R}$", but what exactly is meant by "$\mathbb{R}$"? (i.e. which set theoretic representation ...
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Given any $k\in\Bbb{N}$ there exists $k$ consecutive composite integers such that reciprocal of each composite integer is a Cantor point then $k=1$
Conjecture: Given any $k\in\Bbb{N}$ there exists $k$ consecutive composite integers with the property that reciprocal of each composite integer is Cantor points then $k=1$.
$x$ is a Cantor point iff ...
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Reference for the results of the paper "On a class of probability spaces" by David Blackwell
Is there a modern book that summarizes the results from the paper "On a class of probability spaces" by David Blackwell (1956)?
https://digitalassets.lib.berkeley.edu/math/ucb/text/...
user1123845
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Can a group have $\aleph_1$ many subgroups?
I know that any countable group has either countably many or continuum many subgroups (source), but I'm curious about uncountable groups.
It seems like the proof for countable groups $G$ crucially ...
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Proof of Baire's theorem on functions of first class
I am reading Measure and Category by John Oxtoby, and I have a question about one of the proofs in the book. I reproduce the proof here in full for convenience.
Theorem 7.3. If $f$ can be represented ...
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Infinite vector problem related to the continuum hypothesis
I developed an algorithm that might map the countable ordinal numbers (ones with just a countable number of predecessors) into a subset of the reals without using the axiom of choice. It leads to the ...
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Spaces where each compact subset has compact closure: have they already been studied?
Let $(X, \tau) $ be a topological space be such that for every $K\subset X$ compact implies $\operatorname{cl}(K)$ is also compact.
I am interested in the study of such spaces [call $\textrm{G} $ ...
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On existence of measure-preserving bijection between measurable spaces
Given a collection $\{Q_1,Q_2,\ldots,Q_m\},~Q_i\cap Q_j=\emptyset, i\neq j,~\cup_{j=1}^{m} Q_j = [0,1]$. Define another collection $\{I_1,I_2,\ldots,I_m\}$ such that $\cup_{j=1}^{m} I_j = [0,1]$ and $$...
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G separable group, $\aleph_0 \leq \tau$. What is $l(X)$ and $\omega l(X) (\leq \tau)$? where $X \subseteq G$. And what is $\chi (G)$ (cardinal)?
Happy Chinese new year!
I was reading (and translating) a Russian article "On the topological groups close to being Lindelöf".
Where it is assumed G is a separable group and $\tau \geq \...
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A generalization of Mostowski absoluteness theorem
Studying descriptive set theory, a found the Mostowski absoluteness theorem: if $\phi$ is $\Delta_{1}$, then
$\forall x\in \omega^{\omega} \phi(x)$ is absolute between $V$ and $L$.
I notice that $\...
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Locus of Continuity of a Function between Polish Spaces
I think the question is maybe a bit trivial, but right now my head doesn't work that much well so I'm asking y'all to have some clarification.
I know that, for a subset $A\subseteq \mathbb R$, the ...
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Countable dense subset of Cantor space isn't Wadge reducible to its complement
I am trying to prove that a countable dense subset $Q\subset 2^\omega$ (where $2^\omega$ denotes the Cantor space) is not Wadge reducible to its complement. I am trying to prove this because I want to ...
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Hyperreals and Suslin problem
Hyperreal numbers $^*\mathbb{R}$ form a totally ordered field that is a proper extension of the real numbers (i.e. non-isomorphic to real numbers $\mathbb{R}$). It is obvious that $^*\mathbb{R}$ has ...
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Explicitly constructing a non determined game on $\mathcal{P}(2^{\mathbb{N}})$
I am trying to solve Ex 20.10 on Kechris descriptive set theory book. Throughout this post, a "game on $A$" refers to the infinite game $G(T,X)$ where $T$ is a pruned tree $T\subset A^{<\...
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Descriptive Set Theory - definable sets can be partitioned into more pieces than elements of original set [closed]
I just read that "In Descriptive Set Theory .... definable sets can .... counter intuitively .... be partitioned into more pieces than there are elements in the original set".
Would anyone ...
user239186
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If a metric space $F$ is Borel isomorphic to a Lusin measurable space, then $F$ must be homeomorphic to a Borel subset of some compact metric space
I'm reading about Lusin and Suslin spaces from page 46 of Dellacherie/Meyer's Probabilities and Potential. Here I restrict myself to metric spaces.
A Borel isomorphism between two measurable spaces ...
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How to prove these definitions of an analytic set are equivalent?
Let $F$ be a set and $\mathcal F$ a collection of subsets of $F$ such that $\emptyset \in \mathcal F$. We denote by $F_\sigma$ (resp. $F_\delta$) the closure of $F$ under countable union (resp. ...
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Question on Gale-Stewart Theorem and Axiom of Choice
I am reading Kechris' descriptive set theory text book, and there is this Theorem regarding infinite games:
Gale-Stewart: Let $T$ be a non-empty pruned tree on $A$. Let $X\subset[T]$ be closed or open....
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Proving that the set of points where a continuous function is differentiable is a $\mathbf{\Pi_3^0}$ set
Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function and let $D$ be the set of points where $f$ is differentiable. In order to show that $D\in\mathbf{\Pi_3^0}(\mathbb{R})$, I want to show that $x\...
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Existence of Perfect subset for any given measure
Question: Show that if $0 \leq \text{m}(A) \leq \infty$, then for each positive $q < \text{m}(A)$ there is a perfect set $B \subset A$ of measure $q$.
(All subsets are in $\mathbb{R}$ and the ...
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What is the Lebesgue measure of Luzin's non-Borel set of reals?
On Wikipedia there's a nice example of a non-Borel set due to Luzin. For completeness, I'll summarize it here. For $x\in[0,1]$, let
\begin{align}
x=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 +...
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