The descriptive-set-theory tag has no wiki summary.

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352 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 ...

**4**

votes

**2**answers

91 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 ...

**8**

votes

**2**answers

284 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 ...

**9**

votes

**0**answers

173 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 ...

**7**

votes

**1**answer

140 views

### 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. ...

**16**

votes

**1**answer

461 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 ...

**3**

votes

**1**answer

133 views

### 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 ...

**1**

vote

**2**answers

121 views

### Why the intersection of a scott open (or \w the relatively compactness property) filter on a topology of a sober (and 2nd countable) space is compact?

Definitions and notations.
Let $\mathcal{P}(X)$ the power set of $X$.
Let $\tau_X\subseteq\mathcal{P}(X)$ a topology on X.
We call $A$ irreducible if every time $A=B\cup C$ with $B,C$ closed set ...

**4**

votes

**0**answers

120 views

### 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 ...

**2**

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**0**answers

69 views

### 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. ...

**5**

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**0**answers

207 views

### Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...

**3**

votes

**1**answer

211 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 ...

**2**

votes

**1**answer

188 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 $f\circ g = \mathrm{id}_Y$.
...

**6**

votes

**2**answers

435 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 ...

**4**

votes

**1**answer

109 views

### Analytic enlargement of an analytic set

Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus ...

**8**

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**1**answer

313 views

### Partition $\Bbb{R}$ into a family of sets each one homeomorphic to the Cantor set

It is known that there is no (nontrivial) partition of $\Bbb{R}$ into a countable number of closed set. But is there a partition of $\Bbb{R}$ into sets, each one homeomorphic to the cantor ternary ...

**14**

votes

**2**answers

832 views

### Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets

It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.
For example, ...

**10**

votes

**2**answers

237 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**

votes

**1**answer

172 views

### 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 ...

**10**

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**1**answer

431 views

### 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 ...

**8**

votes

**2**answers

312 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**

votes

**1**answer

279 views

### 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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**1**answer

291 views

### 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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votes

**2**answers

233 views

### 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 $p\in P$ ...

**9**

votes

**1**answer

246 views

### Are there trees for $(\Sigma^2_1)^{\text{uB}}$?

If there is a proper class of Woodin cardinals, then Woodin showed (using stationary towers) that $(\Sigma^2_1)^{\text{uB}}$ statements are generically absolute, where $\text{uB}$ denotes the ...

**29**

votes

**4**answers

7k 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 ...

**5**

votes

**1**answer

381 views

### sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements?
[1] ...

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136 views

### 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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**1**answer

113 views

### ($^{\omega}2$,<) is not well-order. [closed]

Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease ...

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186 views

### $\infty$-Borel Determinacy?

An $\infty$-Borel set is a set $X\subseteq\mathbb{R}$ which has an $\infty$-Borel code - a set $r$ coding the construction of $X$ via open sets, complementation, and well-ordered unions (see ...

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474 views

### Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...

**5**

votes

**1**answer

214 views

### $\omega$ universally Baire sets, tree representations

I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.
...

**9**

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**1**answer

189 views

### Obtaining a lightface pointclass from a boldface one

Define a pointclass to be:
boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...

**5**

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114 views

### A result of Steel on characterizing lightface pointclasses

In the article Projectively wellordered inner models, Steel proves the following theorem (4.12):
Theorem: Let $n < \omega$ and suppose $\mathcal{M}_n^{\sharp}$ exists. Let ...

**4**

votes

**1**answer

220 views

### Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal.
The original proof is in German, and I've been ...

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votes

**1**answer

76 views

### Measurability of $\{ x \in X ; H_0 x \subset A \}$

Let $H$ be some Polish group and $X$ some standard Borel space. Assume that $H$ acts measurably on $X$, i.e. $(h,x) \mapsto hx$ is Borel. Let $H_0 \subset H$ and $A \subset X$ be some Borel sets. Is ...

**5**

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**1**answer

216 views

### A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...

**5**

votes

**1**answer

202 views

### Companion of the pointclass of inductive sets

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures.
I am interested in Spector classes on $\mathbb{R}$, ...

**5**

votes

**3**answers

251 views

### Does every separated measurable space embed into a power of $\{0,1\}$?

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal ...

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258 views

### For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...

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votes

**0**answers

175 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 ...

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**1**answer

156 views

### Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain.
What I need to know is: Can the algebra ...

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votes

**1**answer

252 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 ...

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votes

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183 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 ...

**2**

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**1**answer

338 views

### Descriptive complexity of Hamel bases of R^ω

(base theory = ZFC)
Are any Hamel bases for the vector space $\mathbb{R}^{\omega}$ in the
1. analytical hierarchy?2. projective hierarchy?
In any of the above cases where the answer is not simply ...

**9**

votes

**1**answer

211 views

### 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 ...

**3**

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**0**answers

181 views

### Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...

**0**

votes

**1**answer

69 views

### Existence of a map connecting two marginals of a product measure

Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by ...

**3**

votes

**1**answer

175 views

### Particular neighborhoods of analytical sets

Let $X$ be a standard Borel space: a topological space isomorphic to a Borel subset of a complete separable metric space. Denote by $\mathcal P(X)$ the set of all Borel probability measures over $X$ ...

**6**

votes

**1**answer

388 views

### Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?

Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition:
The meager sets are sets which are ...