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.

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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non-Borel subset of uncountable Tychonoff space

Let $X$ be an uncountable Tychonoff space. Must there exist a non-Borel subset of $X$?
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125 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 $a_{0},a_{1},\dots$ ...
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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 ...
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56 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: ...
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68 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 ...
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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 ...
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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 ...
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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 $\{ ...
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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.
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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 ...
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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) ...
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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 $X$ ...
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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 ...
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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.
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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 ...
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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 ...
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76 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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54 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 $1_s$ ...
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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 $Y \in K \Rightarrow \mathcal F(Y) \subset K$. ...
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78 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 ...
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Cardinality of all $\mathbf{\Sigma}^0_\alpha$-sets over Baire space without full choice

It is well-known that the set of all open (or closed) sets on Baire space has cardinality of the continuum. In context of choice, we can prove that the set of all $\mathbf{\Sigma}^0_\alpha$-sets over ...
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Why do we study Polish spaces?

In descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why ...
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Continuous image of $\mathbb{R}$ is Borel.

Let $X$ be a Polish space and $f\colon\mathbb{R}\to X$. Then since $\mathbb{R}=\bigcup_{n=-\infty }^\infty [n,n+1]$ - a countable union of compact set, $f(X)$ is a countable union of compact sets ...
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Can we construct from $[0,\omega_1)$ a space which is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

I have asked in here a question which tured out to make no sense. I think I have found the confusion and would like to try and rephrase my question: Let $E$ be a topological space, $q \in E$. ...
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Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
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Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)} $ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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In how many steps is it necessary to construct the Borel $\sigma$-algebra?

We only consider Borel sets on $\mathbb R$. As we know, the Borel $\sigma$-algebra is constructed transfinitely as follows: Let $B_0=\mathcal T$ be the set of open sets on $\mathbb R$; If $\alpha$ ...
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Bijection between closed uncountable sets and R? [closed]

This is probably a really stupid question but I'm hoping it's true: can we find a bijection between every closed uncountable set and R (real number line)?
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Polish topological group

A friend asked me to help him prove that the topological group $\mathrm{Homeo}(0,1)$ (homeomorphism of $(0,1)$ with the compact open topology) is Polish (that is, separable and completely metrizable). ...
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Maximal pruned subtree - an absolute notion?

Fix a tree $ p $ over $ \omega $. Let $ [p] $ denote the set of all branches of $ p $. Given a set of reals $ F \subseteq \omega^\omega $, let $ T(F) := \{ x \mathord{\upharpoonright} n : x \in F ...
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homeomorphism of cantor set extends to the plane?

Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ...
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Boolean Closure and Borel sets

Denote the boolean closure of a family of sets $\mathcal S$ by $\mathcal B(\mathcal F)$, then in a metric space it is well known that $\mathcal B(\mathcal F) = \mathcal B(\mathcal G) = \mathcal ...
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On Knaster-Kuratowski fan (aka Cantor's teepee)

As definition of Knaster-Kuratowski fan I take the one on wikipedia. Now, part of the definition of this space is the Cantor set. My question is what particular property of Cantor set we necessary ...
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Measurability of level sets of measures

Let $X$ be a standard Borel space, and $\mathcal P(X)$ be the set of Borel probability measures on $X$ with a topology of weak convergence. It is known that $\{p:p(B) = 1\}$ is a Borel subset of ...
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Measurability of one set of measures

Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ ...
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In $\mathbb{R}$, is a countable union of closed no-where dense sets nowhere dense?

$\mathbb{R}$ is a baire space, so a countable union of closed no-where dense sets: $\bigcup C_n$ has empty interior. That is, $$\bigcup_{A \text{ open }, \; A\subset \bigcup C_n}A = \emptyset $$ So ...