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Results tagged with descriptive-set-theory
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user 141146
Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
2
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112
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Is every Borel function a projection of a Borel function with closed graph?
Is it true the following statement?
Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$, there exists a Polish space $Z$ and a Borel function $g:X \rightarrow Y\times Z$ with closed …
- 2,286
2
votes
1
answer
194
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How much choice is needed to prove this statement?
Consider the following statement (in $\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$):
There exists $(C_\alpha, x_\alpha)_{\alpha \in \omega_1}$ s.t. $C_\alpha \subseteq \mathbb{N}^\mathbb{N}$ is closed …
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1
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0
answers
80
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Separating two sets by a $\boldsymbol{\Delta}_3^0$ set
Let $X$ be a Polish space and $A,B\subseteq X$ be such that $A\cap B = \emptyset$, we know that if there is no $\boldsymbol{\Delta}_2^0$ set separating $A$ from $B$ then there exists a Cantor set $C\s …
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9
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1
answer
702
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How much choice is necessary to prove this statement?
Consider the following statement (in $\mathsf{ZF}+\text{AC}_\omega (\mathbb{R})$):
There exists $(\varphi_\alpha)_{\alpha\in\omega_1}$ with $\varphi_\alpha : \alpha \rightarrow \mathbb{N}^\mathbb{N}$ …
- 2,286
2
votes
1
answer
150
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Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$
My question is:
Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$?
Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\ …
- 2,286
2
votes
1
answer
109
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Strong form of $\mathtt{PSP}$ for $K_\sigma$ sets
Consider an uncountable perfect $K_\sigma$ set $X\subseteq \omega^\omega$, where $K_\sigma$ means countable union of compact sets, perfect means that $X$ has no isolated points and $\omega^\omega$ is …
- 2,286
7
votes
3
answers
345
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Hausdorff quasi-Polish spaces
A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. Pur …
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1
vote
1
answer
125
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Lebesgue Hausdorff Banach theorem for Baire class $1$ functions on $\mathbb{R}^\omega$
A theorem by Lebesgue, Hausdorff and Banach says the following (Kechris' Classical Descriptive Set Theory, p. 192):
Let $X$ be a separable metrizable space and $f: X \rightarrow \mathbb{R}$ be a $\b …
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4
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0
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138
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Separable metrizable spaces far from being completely metrizable
I came across a kind of separable metrizable space that is "far" from being completely metrizable. Before specifying what I mean with "far", I recall that a space is said to be Polish if it's separabl …
- 2,286
3
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1
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188
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Co-analytic $Q$-sets
A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have see …
- 2,286
2
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1
answer
126
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A continuous map relating co-constructible reals
My question is the following:
Given $x,y \in \omega^\omega$ such that $x\equiv_c y$ is there an $L$-definable continuous map $\varphi: \omega^\omega\rightarrow \omega^\omega$ such that $\varphi(x) = …
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3
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0
answers
76
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Forcings that preserve $\mathtt{PSP}$
By $\mathtt{PSP}$ I mean the statement "every subset of the reals has the perfect set property, i.e. either is countable or it contains an homeomorphic copy of the Cantor space $2^\omega$".
I was wand …
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4
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1
answer
229
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Existence of a non-$Q$-set without the perfect set property
We have the following theorem:
Suppose $\omega_1^L=\omega_1$ then there exists a $\Pi_1^1$ subset of reals without the perfect set property
Moreover, under the same hypotheses, we can prove actually …
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3
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1
answer
115
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$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$
Consider the sentence $\mathtt{PSP}_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set".
A priori this sentence is weaker than the usual $\matht …
- 2,286
6
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0
answers
239
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Every Polish space is the image of the Baire space by a continuous and closed map, reference
The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)
Every Polish space (i.e. every separable …
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