# 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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### Existence of a stronger notion of perfect measures

Let $\mathcal{X}$ be a measurable space with its $\sigma$-algebra $\mathcal{B}_\mathcal{X}$ and let $\mathbb{R}$ be the real numbers endowed with its Borel $\sigma$-algebra $\mathcal{B}_\mathbb{R}$. ...
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### Atoms for Markov kernels

Let $X$ and $Y$ be standard Borel measurable spaces. A Markov kernel $f : X \rightsquigarrow Y$ is a map $f(-|-) : \Sigma_Y \times X \to [0,1]$ such that: $f(-|x)$ is a probability measure on $Y$ for ...
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### Intersection of (relativized/preimage) measure 0 with every hyperarithmetic perfect set

Given a perfect tree $T$ on $2^{<\omega}$ viewed as a function from $2^{<\omega}$ to $2^{<\omega}$ define the measure of a subset of $[T]$ to be the measure of it's preimage under the usual ...
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### Complexity of |a| < |b| for ordinal notations?

What is the complexity (e.g. is it $\Sigma^0_1$, arithmetic, fully $\Pi^1_1$) of the relation $|a| < |b|$ given two notations $a, b \in \mathscr{O}$ (Kleene's O)? What about the case where only one ...
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### Does a completely metrizable space admit a compatible metric where all intersections of nested closed balls are non-empty?

(cross-posted from this math.SE question) It is well-known that given a metric space $(X,d)$, the metric is complete if and only if every intersection of nested (i.e. decreasing with respect to ...
315 views

### Is Borel cardinality the same as cardinality under determinacy?

Suppose $E,F$ are Borel equivalence relations on Polish spaces $X,Y$, respectively. Under strong enough determinacy axioms, is it true that $E$ Borel reduces to $F$ iff there is an injective map from ...
431 views

### Can you fit a $G_\delta$ set between these two sets?

Every subset of $\mathbb N \times \mathbb N$ can be viewed as a relation on $\mathbb N$. The set $\mathcal P(\mathbb N \times \mathbb N)$ of all relations on $\mathbb N$ has a natural topology with ...
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### What are the names of the following classes of topological spaces?

The closure of any countable is compact. The closure of any countable is sequentially compact. The closure of any countable is pseudocompact. The closure of any countable is a metric compact set.
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### Upper-bounding determinacy

While the converse of Borel determinacy ("If a set of reals is determined, then it is Borel") is boringly disprovable, I'm curious if there is a sense in which something like it is ...
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### Which topological spaces have a standard Borel $\sigma$-algebra?

Call a topological space $X$ standard Borel if $X$ is standard Borel as a measurable space (equipped with its Borel $\sigma$-algebra), i.e. if there is a Borel isomorphism between $X$ and a Polish ...
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### Does the set of infinite random strings satisfy an analogue of immune sets?

Let $K(x)$ denote the Kolmogorov complexity of a finite binary string $x$. A finite binary string $x$ is called Kolmogorov random if $K(x) \geq |x|$. And an infinite binary sequence is called Martin-...
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### Is there a condensation of a closed subset of $\kappa^\omega$ onto $\kappa^\omega\setminus A$ …?
Let $\aleph_1\le\kappa<c$ and $A\subset \kappa^{\omega}$ such that $\lvert A\rvert\le\kappa$. Is there a condensation (i.e. a bijective continuous mapping) of a closed subset of $\kappa^\omega$ ...