Questions tagged [descriptive-set-theory]

In descriptive set theory we mostly study Polish spaces such as the Baire space, the Cantor space, and the reals. Questions about the Borel hierarchy, the projective hierarchy, Polish spaces, infinite games and determinacy related topics, all fit into this category very well.

Filter by
Sorted by
Tagged with
0
votes
0answers
20 views

How can I show that this set is (or not is) a Borel set?

Suppose we have two Polish spaces $X$ and $Y$ and two Borel injective functions $f\colon X\to Y$ and $g\colon Y\to X$. Consider the following map $$\Phi: \mathcal P(X)\to \mathcal P(Y);\,Z\mapsto X\...
2
votes
0answers
20 views

Dense subsets in the hyperspace of compact sets

Let $X$ be compact and Polish (I'm thinking of $X=[0,1]$, but I guess maybe the same answer holds for every compact Polish space) and let $\mathbf{K}(X)$ be the hyperspace of compact subsets of $X$, ...
3
votes
1answer
31 views

Connectedness and complexity in Polish spaces

I was wondering: How complex can connected subsets of Polish spaces be? Are there connected non-Borel subsets of a Polish space? Given $X$ Polish space (not totally disconnected), does it have proper ...
1
vote
1answer
32 views

Countable union or intersection of Suslin sets is Suslin

We say that a subset of a topological space is Suslin if and only if it is the result of the Suslin operation applied on a Suslin scheme consisting of closed sets. I know that for a Polish space $X$ ...
4
votes
1answer
146 views

Comparing the sizes of null sets

This question is about comparing the relative sizes of null sets by switching from open covers to open covering sequences (a la strong measure zero sets or microscopic sets). The main question is ...
11
votes
1answer
249 views

Could we ever hope to integrate all functions?

The Riemann integral has a weakness, in that it cannot integrate many functions of interest, such as Dirichlet's function $\boldsymbol{1}_\mathbb{Q}$. The Lebesgue and Henstock-Kurzweil integrals ...
1
vote
1answer
30 views

Nomenclature used to refer to the power set $\mathcal{P}$ as a pointclass

This may be silly and vague question, but I'm writing a presentation on the Wadge hierarchy were I refer to different boldface pointclasses , like $\boldsymbol{\Delta}_1^1(X)$ (Borel sets) or $\text{...
0
votes
1answer
58 views

Wadge Hierarchy on $\Delta^{0}_{2}$ sets

i'm studying the Wadge hierarchy on Baire space and Cantor space. I'm asking whether or not the $\Delta^0_2$ sets form a unique degree in these spaces and why the $\Sigma^0_2$-true (i.e. not polish) ...
0
votes
1answer
282 views

Analyzing a definition of average which uses a variation of the Lebesgue integral and measure [closed]

Edit: Why was my post downvoted. Please explain how I can improve my post. Second Edit: I changed my Post Introduction Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. I want to ...
4
votes
1answer
93 views

Problem about $F_\sigma$ and $G_\delta$ sets

Define a set to be bivalent iff it's both $F_\sigma$ and $G_\delta$. Let $X$ be a $G_\delta$-space (i.e. all closed sets are $G_\delta$ sets). Let $G$ and $H$ be disjoint $G_\delta$ sets. Prove ...
1
vote
1answer
37 views

Banach-Mazur game and the Baire property

Given a topological space $\mathcal{X}=(X,\tau)$ and $A\subseteq X$, the Banach-Mazur game on of $A$, $G^{**}(A)$, is the game played as follows: Players $1$ and $2$ alternately play decreasing ...
1
vote
1answer
25 views

Image of Borel set under countable-to-one Borel function is Borel.

Suppose that $ X, Y $ are Polish spaces, $ A \subset X $ is Borel, $ f : X \to Y $ is Borel, and furthermore, $ f $ is countable-to-one: i.e., $ \forall y \in Y: | f^{-1}(y) | \leq \aleph_0 $. Is it ...
3
votes
0answers
74 views

Is the image of Borel measurable function essentially Borel measurable?

It is known that whenever $f: \mathbb{R} \rightarrow \mathbb{R}$ is Borel measurable it is not necessarily true that $f(X)$ is Borel measurable for Borel $X \subset \mathbb{R}$. However I have not ...
2
votes
0answers
29 views

Borel measurable function defined on level sets

Let $(X,\mu)$ be a $\sigma$-finite measure on a standard Borel space $X$ (meaning that $X$ is a separable, complete metric space). Assume that $F:X \rightarrow \mathbb{R}$ and $h:X \rightarrow \mathbb{...
6
votes
1answer
38 views

Showing a particular set is Borel

Let $f:X\times Y \to \mathbb{R}$ be a continuous function, here $X$ and $Y$ are connected and simply connected separable metric spaces. Let $U\subseteq Y$ be non-empty and open. I want to show that ...
4
votes
1answer
86 views

Determinacy and Choice

I know that Kechris proved $$\text{Con}(\text{ZF} + \text{AD})\Rightarrow \text{Con}(\text{ZF} + \text{AD} + \text{DC})$$ And that $\text{DC}$ and $\text{AC}_\omega$ are independent from $\text{AD}$, ...
2
votes
2answers
84 views

Sections of a Borel set in the product space

I have a Polish (completely metrizable and separable) space $X$ and a Borel subet $A \subseteq X\times X$, i.e. $A \in \boldsymbol{\mathcal{B}}(X\times X)$. Given a Borel set $C\subseteq X$ is it true ...
2
votes
1answer
103 views

How do we find the number of sub-intervals whose intersection with the subset of the entire interval is non-empty?

Suppose we have set $A=\left\{\frac{1}{k}:k\in\mathbb{N}\right\}$ and we divide interval $[0,1]$ into $n$ sub-intervals of equal length. How do we find the exact forumula (or approximation) of the ...
1
vote
2answers
36 views

Operation over the Baire space $\omega^\omega$ that preserves Borel sets

We have the following operation on sequences of natural numbers (elements of $\omega^\omega$): $$\begin{align}*:\omega^\omega\times\omega^\omega &\longrightarrow \omega^\omega\\ (x,y) &\...
4
votes
1answer
46 views

Total disconnection and zero-dimension in Polish spaces

First of all Polish spaces are completely-metrizable, separable topological space and by zero-dimensional Polish space I mean that the Polish space has a (countable) basis made of clopen sets. It is ...
4
votes
1answer
50 views

Continuous parametrization of continuous functions in the Baire space $\omega^\omega$

It can be shown that in the Baire space there is a bijection $$\begin{align}\mathcal{N} &\longrightarrow \{f \in {}^\mathcal{N}\mathcal{N} \mid f \text{ is Lipschitz}\}\\ x &\longmapsto \ell_x\...
3
votes
1answer
523 views

Coming up with an equivalent (or close) definition of an average which is easier to compute?

Continuing from my last question, I understand my definition is unclear, I have modified it. I asked the same question on Mathoverflow in case someone there can answer. Definition Consider a ...
3
votes
0answers
75 views

Borel hierarchy within Wadge hierarchy

I'm studying Wadge reducibility and the associated hierarchy (restricted to Borel sets) both from Kechris and more extensive papers. Now, in an exercise Kechris says: $[\emptyset]_W = \{\emptyset\}$...
8
votes
2answers
164 views

There is a $\Sigma^1_1$ universal set and this is not Borel. Where did we use the axiom of choice?

It is well-known that there is a $\boldsymbol{\Sigma}^1_1$ universal set $U \subset \omega^\omega \times \omega^\omega$. That is, there is a $\boldsymbol{\Sigma}^1_1$ subset $U$ of $\omega^\omega \...
2
votes
1answer
55 views

Complexity of formulae involving projective sets in Polish spaces

I am no expert in descriptive set theory, but for some reason want to estimate the complexity of a certain formula. The framework. Let be $F$ a projective set in the Polish space $X=\mathbb R^{\...
4
votes
1answer
71 views

Why does the Cantor-Bendixson cupcake theorem need transfinite induction?

Recall the Cantor-Bendixson theorem: Let $X$ be a Polish space. For every closed subset $K \subseteq X$, there is a unique disjoint sum decomposition $C \cup P = K$ where $C$ is countable and $P$ ...
2
votes
2answers
85 views

Is there a $\mathbb{Q}$-basis of $\mathbb{R}$ that is Borel?

A question that came to my mind yesterday: Applying zorn's lemma (and hence, the axiom of choice), one can show that there must be a Hamel basis for the $\mathbb{Q}$-vector space $\mathbb{R}$, i.e. a ...
0
votes
1answer
40 views

Cantor-Bendixson Theorem — Kechris' proof of perfectness

I am studying Kechris' proof of the Cantor-Bendixson theorem (from his "DST bible", I. 6.B (6.4)). The theorem says the following: Let $(X,\tau)$ be a Polish (i.e. separable, completely metrisable) ...
3
votes
2answers
73 views

Homeomorphic images of an “almost” basic open set in $2^{\omega}$

Let $X\subseteq 2^{\omega}$ such that $X=O\setminus M$ for some basic open set $O$ and a meager set $M$. Suppose that $Y\subseteq 2^{\omega}$ is homeomorphic to $X$. Does it follow that $Y=O'\setminus ...
1
vote
1answer
46 views

Borel $\boldsymbol{\Delta}^0_\alpha$ vs. $\bigcup_{\xi<\alpha}\boldsymbol{\Delta}^0_{\xi}$ for limit $\alpha$

It's clear for limit $\alpha$ that $\bigcup_{\xi<\alpha}\boldsymbol{\Sigma}^0_{\xi}\subsetneq \boldsymbol{\Sigma}^0_{\alpha}$, but is this obviously true for the ambiguous pointclasses? Are there ...
0
votes
0answers
48 views

How can I write a perfect set as $c$ many pairwise disjoint perfect sets

I have been thinking about this question but I have not get complete answer yet. the question is Let $P$ a nonempty perfect subsets of $\mathbb R$ then $P$ can be written as continuum many pairwise ...
4
votes
1answer
94 views

CH implies $\omega_1$ pairwise disjoint of perfect set

This is an exercise in Fundamental of real analysis by James Foran. Prove that the continuum hypothesis implies that each perfect subset of $\mathbb R$ can be written as the union of $\omega_1$ ...
3
votes
1answer
53 views

Writing down a formula for “X is meager” in second order arithmetic with low complexity

So this question of mine arises from the hint of an exercise in Kanamori's "The Higher Infinite", where we try to prove that $\operatorname{Det}(\Pi^1_n)$ implies the Baire property, perfect set ...
4
votes
1answer
46 views

Does proving that closed subset of Polish space is Polish require axiom of countable choice?

Let $C$ be a closed subset of polish space $P$. It is trivial that $C$ is also completely metrizable, but how do we prove that $C$ is separable? I came up with this method: We can prove that separable ...
2
votes
0answers
27 views

Space of positive measures Polish for a weak-star topology?

Let $X$ be a second-countable locally compact Hausdorff space. Denote by $C_c(X) \subseteq C_0(X) \subseteq C_b(X)$ the spaces of continuous functions with compact support, vanishing at infinity and ...
11
votes
1answer
286 views

Is there a specific infinitary sentence second-order logic can't capture?

Below all languages are finite; if preferred, it's enough to work in the language consisting of a single binary relation. By a simple counting argument, there is some $\mathcal{L}_{\omega_1,\omega}$-...
4
votes
0answers
49 views

A step in proving the graph of a Borel map is Borel

In the appendix Takesaki's book on theory of operator algebras, he showed that if $f$ is a Borel map from a metric space $X$ to a metric space $Y$, then the graph of $f$ is a Borel set of $X \times Y$....
2
votes
1answer
84 views

Uncountable closed set $A$, existence of point at which $A$ accumulates “from two sides” of a hyperplane

Let $A \subset \mathbb{R}^d$ ($d \geq 2$) be an uncountable, closed set, such that every point of $A$ is an accumulation point of $A$ ($A$ is perfect). I want to know whether there is some $x_0 \in ...
8
votes
0answers
95 views

Projective conservativity of choice?

Shoenfield's absoluteness theorem says that every $\Sigma^1_3$ statement is upwards absolute between models of ZF with the same ordinals. As a consequence, it implies that ZF and ZFC+V=L prove the ...
2
votes
0answers
34 views

Prove that $\alpha$ is a Borel map

For any $n\ge1$, let us equip the power set of the additive group $(\Bbb{Q}^{n},+)$ with the product, or pointwise, topology induced by the natural bijection between $\mathcal{Pow}(\Bbb{Q}^{n})$ and $...
1
vote
1answer
64 views

Does Borel $\sigma$-algebra cardinality equal cardinality of the continuum?

I know that cardinality of the $2^{\mathbb{R}}$ is greater than cardinality of $\mathbb{R}$. Does the cardinality of the Borel $\sigma$-algebra equal the cardinality of the $\mathbb{R}$?
3
votes
1answer
166 views

Tricky Yet “Elementary” Set Theory Questions

I'm currently working through Jech's Set Theory. It's slow to read but I can understand pretty much everything I come across if I sit down and think about it long enough. However, I don't feel very ...
2
votes
1answer
50 views

Continuous Reduction of $A \subset \mathbb{N}^{\mathbb{N}}$ in $F_{\sigma}$ to countable dense $Q \subset 2^{\mathbb{N}}$.

I am trying to complete problem 21.17 in Kechris' book Classical Descriptive Set Theory, which asks to show that if $Q \subset 2^{\mathbb{N}}$ is countable dense, then $A \leq_{W} Q$ for any $A \...
2
votes
1answer
80 views

Why is $2^{\mathbb{N}}$ Homeomorphic to $2^{\mathbb{N}^{< \mathbb{N}}}$?

This question is based off a problem from Classical Descriptive Set Theory by Kechris. In this book, Kechris makes the claim that, when $\{0, 1\}$ is endowed with the discrete topology, then the ...
0
votes
2answers
85 views

Equivalent of “almost everywhere” for “holds except on a set that is nowhere dense”

When we say that a condition holds everywhere except on a set of measure zero, we can say that the condition holds almost everywhere. I want to say that a condition holds everywhere except on a set ...
1
vote
1answer
40 views

$\Pi_1^1$-complete sets

Let $P \subseteq 2^{\mathbb{N}}$ be a $\Pi_1^1$-complete set. Show that $$R = \{(A,B) \in 2^{\mathbb{N}} \times 2^{\mathbb{N}} : B \subseteq A \, \& \, B \in P\}.$$ is also $\Pi_1^1$-complete. ...
1
vote
1answer
26 views

The Direction of Borel Reducibility

Given that $X$ and $Y$ are Polish spaces and $E$ and $F$ are Borel equivalence relations on $X$ and $Y$, respectively, we say that $E$ is Borel reducible to $F$ if there exists a Borel function $f:X\...
2
votes
1answer
122 views

Subsets of $\mathcal{P}_{\infty}\mathbb{N}$ that are open and dense for the Ellentuck topology are completely Ramsey

I am reading Chapter 10 of Albiac and Kalton's book $\textit{Topics in Banach Space Theory}$, and am trying to understand the proof of Theorem 10.1.3, namely that subsets of $\mathcal{P}_{\infty}\...
1
vote
1answer
44 views

Basic question on complexity of a set

Let $A\subseteq X\times Y$, with $X$ and $Y$ Polish spaces. Suppose that $A=\bigcup_{c\in C}A_{c}$, where $C\subseteq X$ is a closed set and each $A_{c}$ is Borel. Can we conclude that $A$ is ...
1
vote
1answer
46 views

What sets of real numbers are definable in the language of real closed fields?

The language of the first-order theory of real closed fields consists of the non-logical symbols $0$, $1$, $+$, $\cdot$, $<$, and $=$. My question is, for what subsets $X$ of $\mathbb{R}$ does ...

1
2 3 4 5
15