Questions tagged [descriptive-set-theory]

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 descriptive set theory.

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Higher-rank Borel sets

What are interesting, illustrative examples of Borel sets, situated in Borel hierarchy higher than $\Sigma^{0}_{2}$ /$\Pi^{0}_{2}$?
sdcvvc's user avatar
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3 answers
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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 ...
Noah Schweber's user avatar
7 votes
2 answers
780 views

measurable linear functionals are also continuous on separable Banach spaces?

It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.
newbie's user avatar
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7 votes
2 answers
636 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 $\omega^\omega$. ...
Noah Schweber's user avatar
7 votes
1 answer
549 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, ...
Noah Schweber's user avatar
7 votes
1 answer
370 views

What is an example of a meager space X such that X is concentrated on countable dense set?

A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable. What is an example of a separable metrizable (uncountable) meager (...
Alexander Osipov's user avatar
7 votes
2 answers
531 views

Weakly homogeneous trees under AD

If AD$_\mathbb{R}$ holds and $\kappa < \Theta$ then every tree $T$ on $\kappa$ is weakly homogeneous (Martin–Woodin, "Weakly homogeneous trees.") I recall hearing that the hypothesis can be ...
Trevor Wilson's user avatar
7 votes
3 answers
312 views

Hausdorff quasi-Polish spaces

A topological space is said to be quasi-Polish if it is second-countable and completely quasi-metrizable (see for an introduction de Brecht's article: de Brecht, Matthew, Quasi-Polish spaces, Ann. ...
Lorenzo's user avatar
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7 votes
1 answer
588 views

Failure of Shoenfield's Absoluteness

Shoenfield's absoluteness states that if $M \subseteq N$ are models of $ZF$ and $M \supseteq \omega_1^N$, then every $\Sigma^1_2$ formula with parameters in $M$ is absolute between $M$ and $N$. In ...
Ohad Drucker's user avatar
7 votes
1 answer
532 views

Shelah's "Can you take Solovay's inaccessible away?"

I was wandering if there was a book, thesis or some notes where Shelah's argument for $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...
Lorenzo's user avatar
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7 votes
1 answer
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How long can it take to generate a $\sigma$-algebra?

I want to know if there is a $\sigma$-algebra such that for every countable ordinal $\alpha$ the $\sigma$-algebra can be generated in more than $\alpha$ steps but less than $\omega_{1}$ steps. Given ...
Joseph Van Name's user avatar
7 votes
1 answer
250 views

Basis theorem (due to Solovay?)

I'm finishing up my bibliography and I'm looking for a reference for the statement that, working in $L(\mathbb{R})$, the $\Delta^2_1$ sets form a basis for the $\Sigma^2_1$ predicates. I believe that ...
Daniel Walker's user avatar
7 votes
2 answers
737 views

Question about 0-dimensional Polish spaces

Hello everybody, I'm stuck with proving (or disproving) the following statement. Statement: For every $0$-dimensional Polish space $(X,\mathcal{T}\ )$, and a countable basis of clopen sets $\mathcal{...
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7 votes
1 answer
225 views

Weakly homogenously Souslin sets and the measurability of $\omega_1$

I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988): The assertion that every set of reals, in $L(\mathbb{R})$, is the ...
Vincenzo Dimonte's user avatar
7 votes
1 answer
361 views

Iteration of random reals

Consider two random reals $x, y$ over a transitive model $V$ of ZFC. More specifically, if $\mathcal C^V={}^\omega2$ is the Cantor space, composing the canonical homeomorphism with the projections $\...
Carlos's user avatar
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7 votes
1 answer
1k views

Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...
Dabs's user avatar
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1 answer
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Strength of Borel determinacy

In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased). Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $...
new account's user avatar
7 votes
1 answer
476 views

In a Polish space, is every analytic set the continuous image of a Borel set from the same Polish space?

I'm confused by a subtle point in the definition of analytic sets. Suppose I have a Polish space $X$. Now I start with the collection of Borel sets in $X$ and take all their continuous images in $X$. ...
Samuel's user avatar
  • 113
7 votes
1 answer
332 views

Uniformization under AD

Can the following uniformization statement be proved by $ZF+AD+DC$? For any binary relation $R\subseteq \mathbb{R}^2$ with the property that $\forall x (\{y\mid R(x,y)\}\mbox{ is at most countable ...
喻 良's user avatar
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7 votes
2 answers
669 views

Set of perfect subsets of a Borel set

Let $\mathbb{P}$ be the set of all perfect (i.e., every node has incomparable successors) subtrees of the full binary tree $2^{<\omega}$. We can endow $\mathbb{P}$ with a Borel structure by ...
Iian Smythe's user avatar
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7 votes
1 answer
349 views

Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement $\varphi$ ...
Noah Schweber's user avatar
7 votes
3 answers
401 views

Question of combinatorics in the lower part of the Borel hierarchy.

Let $S^\omega$ denote either $\omega^\omega$ or $2^\omega$. Let's call a function $f: S^\omega \rightarrow$ {0,1} 'nice' if there exists a function $g_f: S^{\lt \omega} \rightarrow 2$ such that for ...
Herman Jurjus's user avatar
7 votes
1 answer
280 views

Is there a first-countable space containing a closed discrete subset which is not $G_\delta$?

Being motivated by this problem, I am searching for an example of a first-countable regular topological space $X$ containing a closed discrete subset $D$, which is not $G_\delta$ in $X$. It is easy ...
Taras Banakh's user avatar
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7 votes
2 answers
349 views

Topological spaces with too many open sets

Is there a Tychonoff space $X$ without isolated points with the following property: For any $a\in X$ and any function $f : X\longrightarrow \mathbb{R}$, if $f$ is continuous on $X\backslash \{a\}$ ...
alex alexeq's user avatar
  • 1,851
7 votes
2 answers
484 views

Do continuous maps factor through continuous surjections via Borel maps?

Let $f \colon X \twoheadrightarrow Y$ be a continuous surjection between compact Hausdorff spaces, and $g \colon \mathbb{R} \to Y$ a continuous function. Can you always find a Borel-measurable ...
Chris Heunen's user avatar
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7 votes
1 answer
280 views

Can $\Delta^{1}_{2}$ separate degrees of constructibility?

Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the ...
M Carl's user avatar
  • 437
7 votes
1 answer
203 views

Are σ-sets preserved by Borel isomorphisms?

Recall that a $\sigma$-space is a topological space such that every $F_{\sigma}$-set is $G_{\delta}$-set. $X$ - $\sigma$-set, if $X$ is a $\sigma$-space and it is subset of real line $R$. Let $F$ ...
Alexander Osipov's user avatar
7 votes
2 answers
1k views

Questions on topologies on space of Radon measures

Consider the space $C_c(\mathbb{R})$ of continuous real-valued functions on $\mathbb{R}$ equipped with the inductive limit topology by $C_c(\mathbb{R}) = \bigcup_{n \in \mathbb{N}} C_c(\mathbb{R}, K_n)...
yada's user avatar
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7 votes
1 answer
337 views

Analytic uniformization

Suppose I am given a subset of $2^\omega\times\omega^\omega$ of some bounded Borel rank. Can I get an analytic uniformization of this set?
jonathanverner's user avatar
7 votes
1 answer
208 views

How complicated are 3-player clopen determinacy facts?

Say that a clopen 3-player game is a well-founded tree $T\subseteq\omega^{<\omega}$; intuitively, starting with player $1$ and continuing cyclically, the players $1,2,3$ alternately play natural ...
Noah Schweber's user avatar
7 votes
1 answer
204 views

Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?

Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help). Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
Noah Schweber's user avatar
7 votes
1 answer
416 views

Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal. Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$ ...
Idonknow's user avatar
  • 603
7 votes
1 answer
439 views

Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...
Jefferson Huang's user avatar
7 votes
1 answer
272 views

Kunen tree and Martin tree

Do we know under which conditions the Kunen tree (Recall the Kunen tree provides an analysis of the equivalence classes of functions $f: \omega_1 \to \omega_1$ with respect to the normal measure $W^...
Rachid Atmai's user avatar
  • 3,756
7 votes
1 answer
282 views

Can we inductively define Wadge-well-foundedness?

For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...
Noah Schweber's user avatar
7 votes
1 answer
306 views

On the topological complexity of the set of winning strategies for Gale-Stewart Games

Given a set $A \subseteq \omega^\omega$, let $G_A$ denote the Gale-Stewart game with payoff set $A$ (so player $I$ wants the real built over the course of play to be in $A$ and player $II$ wants it ...
Corey Bacal Switzer's user avatar
7 votes
1 answer
337 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 ...
Cody Dance's user avatar
7 votes
1 answer
466 views

"Robinson arithmetic" for (some) levels of $L$?

I'll write "$\mathcal{L}_\alpha$" for the fragment $\mathcal{L}_{\infty,\omega}\cap L_\alpha$. Say that a countable admissible $\alpha$ is Robinsonian if there is some sentence $\varphi\in\mathcal{L}...
Noah Schweber's user avatar
7 votes
1 answer
293 views

Variously pointed closed sets

A tree $A\subseteq \omega^{<\omega}$ - possibly with dead ends - is pointed iff every path $p\in[A]$ has $p\ge_TA$. This lifts to two distinct notions of pointedness for closed sets in Baire space: ...
Noah Schweber's user avatar
7 votes
1 answer
283 views

Reducing largeness notions, uniformly

This question is a particular take on the following theme. Suppose $A$ and $B$ are two notions of "large subset of $\omega^\omega$;" when is there a uniform method for turning an element of $...
Noah Schweber's user avatar
7 votes
1 answer
227 views

Does a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ have a non-scattered fiber?

Question. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-...
Taras Banakh's user avatar
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7 votes
1 answer
248 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. ...
François Le Maître's user avatar
7 votes
0 answers
365 views

Is Borel cardinality the same as cardinality under determinacy?

Suppose $E,F$ are Borel equivalence relations on Polish spaces $X,Y$, respectively. Under strong enough determinacy axioms, is it true that $E$ Borel reduces to $F$ iff there is an injective map from ...
new account's user avatar
7 votes
0 answers
225 views

Is this equivalent to (some version of) Hechler forcing?

Let $\omega^{<\omega}$ be the set of finite strings of naturals, and let $\omega^{<\omega}_{\not=\emptyset}$ be the set of nonempty finite strings of naturals. Consider the following forcing ...
Noah Schweber's user avatar
7 votes
0 answers
293 views

Which countable ordinals are "Barwise compact" for $\mathcal{L}_{\infty,\omega_1}$?

Barwise compactness says (as a special case) that whenever $\alpha$ is countable and admissible, $T\subseteq\mathcal{L}_{\infty,\omega}\cap L_\alpha$ is $\alpha$-c.e., and every subset of $T$ which is ...
Noah Schweber's user avatar
7 votes
0 answers
453 views

Infinite time Turing machines, semi-decidable sets and descriptive set theory

Definition A set of reals $A$ is said to be ittm-eventually-semi-decidable if there is an Infinite Time Turing Machine programme $P_e$ so that $x\in A$ iff $P_e(x)$ has converged on “1” on its ...
Philip Welch's user avatar
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7 votes
0 answers
143 views

Non-compact dynamical systems

In topological dynamics, most of the time, we consider the continuous action of a (semi)group $G$ on a compact Hausdorff space $X$. In this context, we can envelop the group in a compact left ...
tomasz's user avatar
  • 1,204
7 votes
0 answers
273 views

Co-cones in the Turing degrees

Let the cocone of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $...
Noah Schweber's user avatar
7 votes
0 answers
273 views

A strange planar set and the Continuum Hypothesis

Call a number abnormal if its decimal expansion doesn't feature every digit an infinite number of times. Call a triangle in ${\Bbb R}^2$ abnormal if at least one of its angles spans an abnormal ...
David Feldman's user avatar
7 votes
0 answers
166 views

Choice and the Baire property in non-separable complete metric spaces

It's known to be consistent with ZF+DC that every subset of $\mathbb{R}$ has the Baire property (BP). (E.g. Shelah's model). If so, then every subset of every complete separable metric space has ...
Nate Eldredge's user avatar

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