# Questions tagged [descriptive-set-theory]

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.

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### About perfect sets and perfect Polish space

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### Partitions of the real line into Borel subsets

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### Dependent choices (DC) in ${\bf HOD}(\mathbb{R},X)$, where $X$ is a set of reals

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### A question on Borel equivalence relations

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### Can we have a “very strong” cone phenomenon in the Turing degrees (and a related question)?

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### Scales and concentration

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### Descriptive set theory for computer scientists?

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### Proof of the uniform boundedness theorem for analytic subsets of WO

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### Why measure hyperfinite is equivalent to hyperfinite except for a compressible set?

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### CH and the density topology on $\mathbb{R}$

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### Can there be an upper bound on the Borel rank of the preimages of Borel sets under a surjective Borel map?

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### Can all uncountable (but small) families of sets with positive measure have an uncountable subfamily with an intersection of positive measure?

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### Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$

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### Reference for “$\mathrm{PFA}$ implies $L(\mathbb{R}) \cap \bigcup_{1 \leq k < \omega} \mathcal{P}(\mathbb{R}^k)$ is productive”

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### $f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?

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### What is known about these “explicitly represented” spaces?

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### Non-Borel function from a family of regular Borel measures

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### Borel complexity of special unions of Polish spaces

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### How big is the least non-$\Sigma^1_1$-pointwise-definable ordinal?

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### Can two versions of $\omega_1^{CK}(\mathsf{Ord})$ ever coincide?

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### Is there a Hausdorff space whose “covering problem” has intermediate complexity?

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### Is every element of $\omega_1$ the rank of some Borel set?

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### Incidence relations of subspaces with infinite descending flags

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### Weakly homogenously Souslin sets and the measurability of $\omega_1$

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### How to give a Borel set whose projection is not Borel?

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### Do Borel subsets of the plane with null sections have Borel projections?

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### A rather non-$F_\sigma$ Borel set

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### Sierpinski's characterization of $F_{\sigma\delta}$ spaces

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### What sets can be unraveled?

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### Is the set of all ICC amenable groups countable?

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### Complexity of the set of models of TA

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### Comparing generic versions of $\mathbb{R}$

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### Is the Hilbert cube the countable union of punctiform spaces?

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### Trees of prescribed ordinal

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### Finding 1-generic paths through a tree $T \subseteq 2^{<\omega}$

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### Homeomorphisms and “mod finite”

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### What is the Borel complexity of this set?

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### Non-compact dynamical systems

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### Topological proof that a Vitali set is not Borel

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### Locally compact Polish groups acting on standard Lebesgue spaces

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### Does every cofinal branch through Kleene's O compute true arithmetic?

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### Is there a simple proof that proves $C^1[0, 1]$ is $\Sigma^1_1$ in $C[0, 1]$?

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### Function whose graph is a Borel relation

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### Borel hierarchy and tail sets

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### Lambda system generated by a non-atomic collection

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### Is the co-projection of the projection of a Borel set Lebesgue measurable?

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### Pointwise definable models of determinacy

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### Reference for Borel $\sigma$-algebra of topology of convergence in probability

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### When is validity definable in $L_\alpha$?

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