**1**

vote

**0**answers

33 views

### Limit inferior of Borel functions [on hold]

Suppose $X$ is separabile metric and $F \colon X \times \mathbb{R}_+ \to [ 0 , 1 ]$ is Borel. Let $ f ( x ) = \liminf_{\varepsilon \to 0} F ( x , \varepsilon )$.
Is $f$ Borel?

**1**

vote

**0**answers

81 views

### Measurable selections of a finite familiy of measures

EDIT. I'm adding a missing hypothesis and a really TL;DR version of the core problem. Warning: This short statement is the strongest form of what I want, hence not as plausible as the original form.
...

**7**

votes

**2**answers

285 views

### Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...

**11**

votes

**1**answer

304 views

### Non meager rectangle

Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?

**5**

votes

**1**answer

186 views

### Proof of a soft version of Moschovakis's lemma

The following fact, which I've heard being called "soft version of Moschovakis's lemma" (see top answer here) is the following:
Under AD, if there is a surjection $\Bbb R\rightarrow\alpha$, then ...

**7**

votes

**2**answers

506 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 ...

**9**

votes

**1**answer

379 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...

**9**

votes

**0**answers

181 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...

**2**

votes

**0**answers

200 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**5**

votes

**0**answers

139 views

### Analytic equivalence relations whose classes are sometimes Borel

There are analytic equivalence relations for which the statement "All classes are Borel" is independent of $ZFC$. In all the examples I know about, the classes are non Borel in $L$ or $L[z]$ for some ...

**5**

votes

**0**answers

85 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 ...

**5**

votes

**1**answer

204 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 ...

**-1**

votes

**1**answer

109 views

### Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...

**9**

votes

**1**answer

284 views

### Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$.
Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...

**5**

votes

**1**answer

176 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 ...

**6**

votes

**0**answers

255 views

### Reference for “if the set $A$ is Suslin, then every $\Sigma^1_1(A)$ set is Suslin”

Does anyone know of a reference for one or both of the following facts (in $\mathsf{ZF}$)?
If the set of reals $A$ is Suslin, then every $\Sigma^1_1(A)$ set of reals is Suslin.
If $T$ is a tree on ...

**9**

votes

**2**answers

189 views

### Descriptive Complexity of Knot Equivalence

I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete ...

**3**

votes

**1**answer

53 views

### Illfounded trees as “retract” of all trees

Definitions: Let $\omega^{<\omega}$ be the set of all finite sequences of natural numbers. For $u, v \in \omega^{<\omega}$, let $u \prec v$ denote that $u$ is a prefix of $v$. We call a subset ...

**7**

votes

**1**answer

182 views

### Characterizing L(R) Cardinals in HOD

We're working in L(R) under AD.
We know that
$\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc.
My question is about ...

**3**

votes

**0**answers

45 views

### If $x≤^∗y$ has the $\text{BP}$(in $X^2$) we can say that A is meager?

Let $X$ be a Polish space. Let $(I,<)$ be a wellordered set and $(A_i)_{i∈I}$ a family of sets meager in $X$. Let $A=⋃_{i∈I}A_i$. If consider the relation $x≤^∗y$ defined by:
$x,y∈A$ $\wedge$ (the ...

**3**

votes

**0**answers

98 views

### Fixed Points of the Friedman Stanley Jump

Consider the situation of a pair $(X,E)$, where $X$ is a standard Borel space and $E$ is an invariant equivalence relation on $X$*. The Friedman-Stanley jump of this pair is an equivalence relation ...

**6**

votes

**1**answer

176 views

### Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code

It is well known that every Borel set has the property of Baire. That is, for every Borel set $B$, there is an open set $U$ and a sequence of dense open sets $D_n$ such that for every $x\in \cap_n ...

**2**

votes

**1**answer

122 views

### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...

**5**

votes

**1**answer

297 views

### A property of the Frechet filter and every ultrafilter

(Joint question with Piotr Szewczak.)
Definitions and notation. By filter we mean a filter on $\omega$ containing the cofinite sets at least.
For a filter $\mathcal{F}$, let ...

**11**

votes

**1**answer

279 views

### A classic cardinal characteristic of the continuum in disguise?

We believe the answer to the following question, that is relevant to a joint research project with Piotr Szewczak, should be known. We would appreciate any help or pointer.
Needed definitions may be ...

**1**

vote

**0**answers

104 views

### Topological properties of space of Radon measures

Let $M$ denote the space of signed unbounded Radon measures on $\mathbb{R}$ as is defined by Bourbaki, i.e. $M$ is the dual of $C_c$ where $C_c$ is the space of continuous functions on $\mathbb{R}$ ...

**6**

votes

**1**answer

186 views

### Surjective (strong) reducibility of Borel equivalence relations

Suppose $E$ and $F$ are Borel equivalence relations on Polish spaces $X$, $Y$, resp. Say that $E$ is surjectively Borel reducible to $F$ iff there is a Borel surjection $f:X \to Y$ such that $xEy$ iff ...

**2**

votes

**0**answers

290 views

### A query on how to climb inaccessibles in £

I am investigating to what extent extensions of the librationist property or set theory £ may support relative inaccessible sets; see Librationist Closures of the Paradoxes and Elements of ...

**6**

votes

**3**answers

259 views

### Borel cross section

It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...

**16**

votes

**6**answers

873 views

### Application of Fraïssé construction in set theory

As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.
Now I would like to know ...

**3**

votes

**1**answer

159 views

### Countable chain condition in $\text{BP}(X)$

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$.
Assume $X$ is second countable Baire ...

**2**

votes

**0**answers

68 views

### Potentiality classes and Borel reductions

In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ ...

**3**

votes

**1**answer

200 views

### Question about of comeager set

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...

**7**

votes

**0**answers

130 views

### A jump operator for Borel equivalence relations

It is well-known that with respect to Borel reducibility the class of Borel equivalence relations on a standard Borel space does not admit a maximal element. We can use the well-known Friedman-Staley ...

**4**

votes

**1**answer

131 views

### Does a surjective measurable map induce a surjective pushforward operator?

I hope it is OK to post a question that is basically the same as the months old currently unanswered question at math stackexchange
Suppose X, Y are Polish spaces (without loss of generality, we may ...

**7**

votes

**1**answer

190 views

### Defining functions pointwise vs. almost everywhere (w.r.t. uncountably many mutually singular measures)

My question is motivated by a general measure-theoretic problem that one frequently encounters in probability: the need to work with uncountably many mutually singular measures at once, and with ...

**9**

votes

**0**answers

221 views

### Proving regularity properties from forcing axioms

It's well known that PFA implies projective determinacy. It's also well known that PD implies that all projective sets are Lebesgue measurable, have the Baire property, etc.
Is there a direct proof ...

**4**

votes

**2**answers

285 views

### Limits of determinacy on reals

For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built ...

**1**

vote

**1**answer

162 views

### A question regarding sets of Vitali's type in models of $ZF+GCH$ where $L$$\neq$$V$

Consider sets of Vitali's type in models of $ZF+GCH$ where $L$ $\neq$$V$. Are there sets of Vitali's type in both $L$ and $V-L$? If so, is there any way one can distinguish the constructible sets of ...

**5**

votes

**0**answers

261 views

### Cardinal characteristics without choice

(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...

**2**

votes

**0**answers

126 views

### Classify spaces that make extension theorems hold

Recall a Polish space is a completely metrizable separable space.
Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...

**7**

votes

**2**answers

356 views

### When can we reach a real by forcing?

I'm sure this is well-known, but: suppose I have a non-constructible real $r\in V-L$. Under what conditions is there a poset $\mathbb{P}\in L$ and a $G$ which is $\mathbb{P}$-generic over $L$, such ...

**1**

vote

**1**answer

225 views

### At what level of the analytic hierarchy do Cohen reals lie?

In his doctoral thesis titled "Three models of ordinal computability", Benjamin Seyfferth proved the following theorems:
i) A set $\mathtt A$ of reals is Ordinal Turing Machine-enumerable if and only ...

**7**

votes

**1**answer

179 views

### Space of Borel measurable maps

That's a question from MSE (here) that did not receive any answer for some days. I migrate it to MO.
Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f: ...

**1**

vote

**1**answer

202 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 ...

**10**

votes

**3**answers

496 views

### 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 ...

**9**

votes

**0**answers

217 views

### 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 ...

**8**

votes

**2**answers

346 views

### 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 ...

**5**

votes

**3**answers

140 views

### 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 ...

**16**

votes

**1**answer

652 views

### 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 ...