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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Concerning the various proofs from the axiom of choice that R^3 admits of surprising geometrical decompositions into circles, skew lines and so on: can we prove in any instance that there are no Borel such decompositions? Or that AC is required?

This question follows up on a comment I made on Joseph O'Rourke's recent question, one of several questions here on mathoverflow concerning surprising geometric partitions of space using the axiom of ...
54
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2answers
5k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
39
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4answers
10k views

Non-Borel sets without axiom of choice

This is a simple doubt of mine about the basics of measure theory, which should be easy for the logicians to answer. The example I know of non Borel sets would be a Hamel basis, which needs axiom of ...
11
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2answers
288 views

Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
3
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1answer
538 views

Different Metrics for Baire Space and their induced Topologies

The Baire-Space is the set of all infinite sequences of integers, i.e. $$ \mathcal N = \omega^{\omega}. $$ On this space usually the following metric is given $$ d(\alpha, \beta) = \left\{ ...
6
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1answer
92 views

Can each non-open analytic subgroup of a Polish abelian group be covered by countably many closed Haar null subsets?

By a result of Laczkovich ('Analytic subgroups of the reals' Proc AMS Vol 126 (1998)), any non-open analytic subgroup of a Polish locally compact group can be covered by countably many closed Haar ...
3
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1answer
329 views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation ...
14
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3answers
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Parts of Set Theory immune to independence

The motivation for asking this question is a passage (3.2) in an article by Greg Hjorth where he said that "...it is also an attractive feature of the theory of Borel cardinalities and of the theory ...
16
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1answer
473 views

Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering ...
9
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1answer
259 views

Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre. Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...
7
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2answers
562 views

Avoiding countable subgroups of general uncountable groups

The following problem is a general form of another problem (motivation is available there). Initially, the problems were posted together, but the first one is solved below, a solution that does not ...
11
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1answer
299 views

Which forcings preserve (some) determinacy?

The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
11
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0answers
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Absoluteness of “$\kappa$-homogeneously Suslin” for sets of reals

What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals? For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > ...
6
votes
2answers
580 views

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 ...
5
votes
1answer
200 views

Universally Baire Tree Representation of Projective Sets

In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
4
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1answer
91 views

On continuous perturbations of functions of the first Baire class on the Cantor set

Is it true that for any function of the first Baire class $f:X\to\mathbb R$ on the Cantor cube $X=2^\omega$ there is a continuous function $g:X\to[0,1]$ such that the image $(f+g)(X)$ is disjoint with ...
11
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2answers
1k views

Generalizations of the Tietze extension theorem (and Lusin's theorem)

I am reasking a year-old math.stackexchange.com question asked by someone else. (For my needs every space $X$ and $Y$ will be Polish---that is a completely separably metrizable space.) The Tietze ...
10
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1answer
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Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?

Is it consistent with ZFC that there exists an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set? If the continuum hypothesis holds, or more generally ...
10
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1answer
483 views

Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
6
votes
1answer
236 views

$\text{cov}(\mathcal{M})$ vs. $\mathfrak{b}$ vs. $\mathfrak{s}$

Let me first recall some pretty standard notations: $\text{cov}(\mathcal{M})$ is the covering number of the ideal $\mathcal{M}$ of all meager subsets of $\mathbb{R}$; $\mathfrak{b}$ is the bounding ...
4
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1answer
277 views

Indeterminacy of long games

Hello, all, Several months ago I sat in on a seminar on AD+, which was incredibly wonderful even though I could barely follow it at all. AD+ is a technical variant of AD, the axiom of determinacy, ...
2
votes
1answer
226 views

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 $f\circ g = \mathrm{id}_Y$. ...
9
votes
2answers
352 views

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 ...
8
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1answer
287 views

Can $\mathbb{R}$ be partitioned into dedekind-finite sets?

Assuming $ZF$ itself is consistent, it is consistent that there are sets $D$ which are infinite but cannot be placed in bijection with any of their proper subsets; such sets are called "strictly ...
4
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2answers
239 views

Natural examples of $\bf\Sigma^0_3$ equivalence relations

I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than ...
4
votes
1answer
113 views

Is it possible for a separable metric and a non-separable metric to have the same Borel $\sigma$-algebra?

I hope this is not too basic or obvious a question. Let $d_1$ and $d_2$ be metrics on the same set $X$, with $d_1$ being separable and $d_2$ not being separable. Is it possible that $d_1$ and $d_2$ ...
4
votes
1answer
238 views

Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]

Definition: Suppose $\mathcal A$ is the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and ...
3
votes
0answers
324 views

“Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero". A set $X$ of reals is strong measure zero if, for any $f: ...
3
votes
0answers
216 views

How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?

Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets: (1) What ...
1
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1answer
208 views

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 ...