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.

learn more… | top users | synonyms

6
votes
1answer
121 views

Turing invariance on large sets

Definition: A function $f: 2^{\omega} \rightarrow 2^{\omega}$ is Turing invariant if $x \equiv_T y \rightarrow f(x)\equiv_T f(y)$. Question I (under $ZFC$): Let $f: 2^{\omega} \rightarrow 2^{\omega}$ ...
5
votes
1answer
73 views

Separating disjoint sets of size $\aleph_1$ with Borel sets

Question: Is it consistent with ZFC that every pair of disjoint sets $A,B\subseteq\mathbb{R}$, both of size $\aleph_1$, can be separated by a Borel set? This statement is clearly false under CH; take ...
1
vote
0answers
25 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
2
votes
1answer
47 views

Cardinality of a set of positive Lebesgue measure

I have pretty no knowledge in set theory, so likely the question has a trivial answer. All countable subsets of $[0,1]$ have Lebesgue measure of zero, thus all sets of positive Lebesgue measure are ...
3
votes
0answers
21 views

Polish subspaces of the space of functions into a Polish space

For a topological space $E$ denote $\mathscr{B}(E)$ its Borel $\sigma$-algebra. For a measurable space $(A, \mathscr{A})$ and $B \subseteq A$ denote $\mathscr{A}|B$ as the trace $\sigma$-algebra on ...
2
votes
1answer
32 views

No-where dense sets in the reals

We have a theorem $A\subseteq \mathbb{R}$ is n.w.d. $\longleftrightarrow \text{cl}(A)$ is n.w.d. Where n.w.d. means "no-where dense" of course... I was thinking to myself-- does there exist two ...
0
votes
0answers
27 views

Countable metric space and space of all rational numbers.

I know that uncountable metric space can not be homeomorphic to a subspace of the space of $\mathbb{Q}$ of all rational numbers. but can this be true for countable metric space?
1
vote
1answer
28 views

Borel subset in Polish Spaces

Let $f:X\rightarrow Y$ be a continuous onto mapping between Two polish spaces,then are these two statements true:- Image under $f$ of the every Borel subset in $X$ is a Borel subset in $Y$. Inverse ...
2
votes
2answers
46 views

Intuitive way to understand the Smith–Volterra–Cantor set

The Smith–Volterra–Cantor set (or ε-Cantor set) is a set of points on $ℝ$ that is nowhere dense, yet has positive measure. As I understand it, being nowhere dense means containing no intervals. An ...
3
votes
1answer
43 views

Continuum Hypothesis for closed sets

In A Beginner's Guide to Modern Set Theory [page 48], the author says: [Cantor] did prove that every closed uncountable subset of $\mathbb R$ has cardinality $2^{\aleph_0}$... ... but I cannot ...
2
votes
2answers
126 views

Open set of Polish space is again a Polish space?

A Polish space ​is a separable completely metrizable topological space. On the wikipidia article and in the book measure theory from Bauer (§26, Example 4) is stated that any open set of a polish ...
3
votes
3answers
77 views

Cardinality of non-Borel sets

Assume ZFC. Let $B\subseteq\mathbb R$ be a set that is not Borel-measurable. Clearly, $B$ must be uncountable, since countable sets are always Borel being a countable union of measurable singletons. ...
2
votes
0answers
26 views

Space of Borel measurable maps

Let $X$ and $Y$ be two standard Borel spaces and consider the set $M(X,Y)$ of measurable maps $f : X \to Y$. Is $M(X,Y)$ also standard Borel? First of all, the cardinality of $M(X,Y)$ is ...
1
vote
1answer
46 views

Is a function with everywhere discontinuities of the second kind always measurable?

Let $f : [0,1] \to \left\{ 0, 1 \right\}$ be a function that has at each point a discontinuity of the second kind. Is $f$ measurable if we equip the domain with the Borel or even Lebesgue ...
4
votes
0answers
46 views

Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
1
vote
2answers
54 views

Continuity of a mapping $C\to C^2$, $C$ being the Cantor set

I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$. Now we consider ...
1
vote
0answers
37 views

Is there a model of set theory in which $2^{2^{\omega_1}}$ is separable?

We know that $2^{\mathfrak c^+}$ ($\mathfrak c =2^\omega=|\mathcal P (\omega)|$) is not separable by the following argument: Suppose $D$ is countable dense in $2^{\mathfrak c^+}$. For each ...
1
vote
1answer
46 views

Lightface Pointclasses

I know that a predicate $A\subset\mathcal{N}$ is said to be $\Sigma^0_1$ if there is a recursive ($\Delta^0_1$) monotone predicate $R\subset\omega$ s.t. $A(f)\iff\exists nR(f\restriction n)$. Here ...
0
votes
1answer
41 views

Every analytic subset of $\Bbb{R}$ is the projection of a $G_\delta$ set $G \subset \Bbb{R} \times \Bbb{R}$

In the answer to this question (Projection of a set $G_\delta$, respectively in this post http://mathoverflow.net/questions/34142/projection-of-borel-set-from-r2-to-r1) it is claimed that every ...
1
vote
1answer
49 views

If $\beth_1$ is weakly inaccessible, are any of the cardinal characteristics of continuum provably strictly less than $\beth_1$?

Assume ZFC+"$\beth_1$ is weakly inaccessible." Are there any cardinal characteristics of the continuum mentioned at wikipedia that can thereby be proved to have cardinality strictly less than ...
1
vote
1answer
50 views

Projection of a set $G_\delta$

The canonical projection $\pi:\mathbb{R}^2\rightarrow \mathbb{R}$ such that $\pi(x,y)=x$ maps $G_\delta$ sets to Borel sets? i.e. If $A=\cap_n^\infty A_n$ with $A_n$ open sets, then $\pi(A)$ is ...
4
votes
1answer
98 views

How many subspace topologies of $\mathbb{R}$?

Say two subsets of $\mathbb{R}$ are equivalent if they are homeomorphic, with the subspace topology. How many equivalence classes are there? It's immediate that there are at least $\beth_0$ (we can ...
6
votes
1answer
95 views

Probability that an element belongs to an infinite subset of a set

$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an ...
3
votes
2answers
84 views

The Cantor set is not strong measure zero

$A \subseteq \mathbb R$ is strong measure zero if given any sequence $( \epsilon_n )_{n \in \mathbb N}$ of positive reals there is a sequence $( I_n )_{n \in \mathbb N}$ of open intervals such that ...
1
vote
1answer
44 views

Does $X \subseteq \mathbb N^{\mathbb N}$ non-countable and $F_{\sigma}$ imply that $X$ contains a perfect set?

I think that the claim below is true, but whenever I try to prove it, I find myself using the continuum hypothesis ($\aleph_1 = \mathfrak c$). My question: Can the following statment be proved ...
3
votes
3answers
92 views

non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
7
votes
1answer
128 views

All games determined + ZF inconsistent

Let $A$ be a nonempty set, $T\subset A^\mathbb{N}$ a nonempty pruned tree and $X\subset [T]$. The game $G_{A}(T,X)$ is played as follows: Player I and Player II take turns playing $a_{0},a_{1},\dots$ ...
3
votes
0answers
31 views

pseudo inverse of a finite-to-one continuous map and measurability

Given that $\pi: X \to Y$ is a continuous onto map between compact metric spaces such that the fiber $\pi^{-1}(y)$ is a finite subset of $X$ for all $y$, is the map $y \mapsto \pi^{-1}(y)$ guaranteed ...
2
votes
1answer
59 views

Is it true that $A$ is scattered?

Let $X$ be a (Hausdorff) topological space and for each ordinal $\alpha$ denote by $X^{(\alpha)}$ the $\alpha$th derivative of $X$ by the Cantor-Bendixson derivation (i.e., define transfinitely: ...
2
votes
1answer
69 views

Closure of $\mathbf{\Sigma_{n}^{0}} $ under finite cartesian products

An exercise in Moschovakis' descriptive set theory, asks to show if we have pointsets $P\subset\mathcal{X}$ and $Q\subset\mathcal{Y}$ both of which are $\mathbf{\Sigma_{n}^{0}}$, then to show that ...
10
votes
1answer
116 views

Partitions of the Cantor space into parities

Call a partition of $2^\mathbb{N} = A\cup B$ a parity partition if, for any $n\in\mathbb{N}$, flipping the $n$th bit of any element of $A$ results in an element of $B$, and vice-versa. Given a choice ...
2
votes
1answer
35 views

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets? (or equivalently, that $\mathbb R$ contains no more then $\mathfrak c$ $G_\sigma$ sets? The more general ...
4
votes
0answers
38 views

The principle $S_1(\mathcal O,\mathcal O)$ versus the game $G_1(\mathcal O,\mathcal O)$

Given a topological space $X$, Let $\mathcal O$, denote the set of all open covers of $X$. We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ ...
1
vote
1answer
107 views

Reference request, Descriptive set theory

I was wondering what a good text would be to learn descriptive set theory out of? Hopefully something more in the spirit of Kunen's text on the introduction to independence proofs.
2
votes
1answer
68 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
0
votes
0answers
25 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
3
votes
1answer
40 views

Is the set of continuous function with Lebesgue zero set a Borel set in continuous space?

Let $D$ be a domain in $\mathbb{R^d}$ and denote the continuous function space on $D$ as $X := C(\overline{D})$ where we can define the $\sigma$-algebra $\mathscr{B}(X)$ of $X$, that is sets in $X$ ...
2
votes
0answers
25 views

Strong Choquet preimage implies strong Choquet?

Recall that a strong Choquet space is one where player II has a winning strategy in the game where two players take turns: player I chooses an open set and a point inside, then player II chooses a ...
1
vote
0answers
23 views

F sigma subsets of the real line

I need an example of a nowhere dense $\mathcal F_\sigma$ subset of the real line that is not a countable union of perfects sets and not a countable union of pairwise disjoint closed sets.
1
vote
1answer
49 views

The Cantor set and it's connection to t(C_p(X))

I am reading the following proof (from page 124 here) which is a part of a larger proof for the statement: Exercise 151: Show that there exist spaces $X$ and $Y$ such that, $t(C_p(X)) = \omega$ and ...
1
vote
0answers
25 views

Complexity of a Borel linear subspace of a Banach space

This question is inspired by the MO question Image of $L^1$ under the Fourier transform, but I think it might be much easier so I am posting it here instead. Let $(X, \|\cdot\|)$ be a separable ...
3
votes
1answer
79 views

Conditional probability independent of one variable

Let $X,Y$ and $Z$ be Borel spaces (that is, Borel subsets of Polish spaces) and let $\mathcal P(X)$ denote the Borel space of all Borel probability measures on $X$. For a product measure $P\in ...
1
vote
2answers
73 views

Homeomorphism between $\prod X_i$ and the Cantor set

I've been trying to find a homeomorphism $$\Phi:\prod_{i=1}^\infty X_i \to \{ 0, 1\}^\mathbb{N},$$ where each $X_i$ is a finite set with at least two elements, but have been unable to. Things I've ...
3
votes
0answers
38 views

How to show: a non-scattered Čech-complete space, contains a compact subspace which can be continuously mapped onto the Cantor-set

It is written in this article that "a non-scattered Čech-complete space, contains a compact subspace which can be continuously mapped onto the Cantor-set". Can anyone explain this? I mean, if, $X$ is ...
4
votes
2answers
121 views

Banach-Mazur Game: Proof about winning strategies

I have to hold a presentation about the Banach-Mazur-Game to undergraduates this week. It should all stay very simple, so I will mainly only talk about the "original" Banach-Mazur Game on ...
4
votes
2answers
109 views

Cardinality of an algebra

Suppose that $B$ is the Boolean algebra of all Lebesgue measurable sets in $I=[0,1]$ modulo Null sets. I am asking (1) What will be the cardinality of $B$. Does it have to be $|B|=\mathfrak ...
2
votes
1answer
56 views

Non analytic sets?

I'm studying Jech's Set Theory and I noticed that he doesn't show an example of a non analytic set. Of course the analytic sets are Lebesgue measurable, have the Baire property and the perfect set ...
0
votes
1answer
57 views

Infinite product space (TOPOLOGY)

I am asking about the infinite product of $\{0,1\}$. That is, $\{0,1\}^\Bbb N$ is the space of all infinite sequence of $0_s$ and $1_s$, how collection of all finite sequences of $0_s$ and $1_s$ ...
1
vote
0answers
25 views

The game $G(K,X)$

In Telgarsky - Topological games, in page 246, the following game $G(K,X)$ is described: There are given a space $X$ and a class $K$ of spaces such that $Y \in K \Rightarrow \mathcal F(Y) \subset K$. ...
1
vote
1answer
86 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...