# Tagged Questions

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 ...

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### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1]. Germs of analytic functions can be distinguished by derivatives at a point. So in both cases we see ...
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### The continuum hypothesis for packing shapes without overlapping

Consider the finite cross $C$ (=union of line segments $\overline{(0, -1)(0, 1)}$ and $\overline{(-1, 0)(1, 0)}$) and the unit half-circle $H$. It is easy to see that we may pack continuum-many ...
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### Is $\mathbb{Z}^{\omega}$ ever the union of a chain of proper subgroups each isomorphic to $\mathbb{Z}^{\omega}$?

Recall that the covering number $cov(B)$ is the least cardinal $\kappa$ such that $\kappa$ meagre sets cover the real line. Andreas Blass and John Irwin http://www.math.lsa.umich.edu/~ablass/bb.pdf ...
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### cardinality of perfect sets in generalized Baire space

I've been unable to find an answer to the following question in the literature on generalized descriptive set theory. Consider Baire space $\kappa^{\kappa}$ where $\kappa$ is inaccessible. The basic ...
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### Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...
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### Three old questions on the Sacks forcing

I came across the two following Qs in 1970. Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
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### The (global) theory of Borel equivalence relations

What do we know about the complexity of the theory of Borel equivalence relations, with the Borel reducibility order $\leq_B$? That is, let $\mathcal{B}$ be the set of all Borel equivalence ...
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### Uniformization/measurable selection theorems

Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. ...
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### Inverse of a Borel surjection

Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$. ...
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### Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
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### Countable model theory for $\omega$-stable theories?

This is a bit of a fishing expedition, because I'm not sure what I'm looking for. Very vaguely stated, here's the driving question: What conditions on an $\omega$-stable theory make the class of ...
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### Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$

The following excerpt is from Connes' Noncommutative Geometry Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel ...
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### Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
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### When is $A$ “$L$-ish” whenever $B$ is “$L$-ish”?

My question is about a kind of relative constructibility in set theory. Fix a countable transitive model $W\models ZFC$ which is much bigger than $L^W$. There is a natural way within $W$ to compare ...
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### Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
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### Is the set of measurable maps with countable range Borel?

Let $(X,\mu)$ be a standard probability space, and $(Y,\tau)$ an uncountable Polish space. Then the set $L^0(X,\mu,Y)$ of measurable maps from $X$ to $Y$ identified up to measure 0 is Polish w.r.t. ...
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### Higher recursion theory and reverse mathematics: What is to $\Pi^1_1-CA_0$ as $RCA_0$ is to $ACA_0$?

There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" - indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in ...
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### Cofinality of a $\sigma$-ideal of $\mathbb{R}$

The cofinality of a partially ordered set $\left( P,\leq \right)$, written $cof(P)$, is the smallest cardinality of a subset $T$ of $P$ that is [EDIT: cofinal] in $P$, i.e. for every element $p\in P$ ...
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### Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...
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