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**47**

votes

**2**answers

3k 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 ...

**27**

votes

**0**answers

989 views

### 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 ...

**10**

votes

**2**answers

245 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 ...

**2**

votes

**1**answer

389 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\{ ...

**33**

votes

**4**answers

8k 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 ...

**3**

votes

**1**answer

238 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**

votes

**3**answers

1k views

### 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 ...

**11**

votes

**1**answer

249 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 ...

**10**

votes

**1**answer

426 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 ...

**5**

votes

**2**answers

467 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 ...

**2**

votes

**1**answer

203 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

**2**answers

334 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 ...

**9**

votes

**1**answer

818 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 ...

**8**

votes

**1**answer

263 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**

votes

**1**answer

201 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

**1**answer

236 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

**0**answers

180 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**

vote

**1**answer

199 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 ...