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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What is the definition of the following concepts and how I can characterize each concept. [closed]

What is the definition of the following concepts and how I can characterize each concept. A set $A\subseteq 2^{\omega}$ is Lebesgue measurable zero if ? A set $A\subseteq \omega^{\omega}$ is ...
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1answer
26 views

Product of Borel and non-Borel set

It is true that product of Borel and non-Borel sets is non-Borel set? More precisely, I would like to know if $V $×$ \{1\}$ is Borel, where $V$ is Vitali set.
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279 views

Continuous surjections onto $\mathbb{R}$

I have two questions about continuous functions: Suppose $X \subseteq \mathbb{R}$ and $X$ has same cardinality as $\mathbb{R}$. Can we find a continuous function from $X$ onto $\mathbb{R}$? Suppose ...
5
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2answers
111 views

A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
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1answer
49 views

Continuous function on Polish space

I want to prove the following and have no idea how to proceed: For a continuous function $f: X \mapsto Y$ where $X$ is Polish and $Y$ is Hausdorff the following are equivalent: $f[X]$ is ...
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1answer
30 views

Is the Borel-sigma-Algebra of a Polish space always countably generated?

Wikipedia says: Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure. (Polish space, Wikipedia) So since the ...
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1answer
29 views

Is the Borel reduction of an hyperfinite equivalence relation still hyperfinite?

Let $(X,E)$, $(Y,F)$ be Borel equivalence relations. Assume $(X,E)\leq_{\text{B}}(Y,F)$.${}^{\dagger}$ Does it hold that: $F$ hyperfinite $\Longrightarrow$ $E$ hyperfinite?${}^{\ddagger}$ I do not see ...
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42 views

Undetermined game of length $\omega_1+\omega$, without choice

On the following page, Taranovsky is talking about his "Determinacy Maximum" axiom: http://web.mit.edu/dmytro/www/DeterminacyMaximum.htm He also justifies the choice of the name, by pointing out that ...
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1answer
61 views

Not meager in $V^{\mathbb{C}_I}$

Assume $A\subseteq 2^\omega $ is not meager in any non-empty open set, in The ground model $V $. Then is not meager in any non-empty open set, in $ V^{\mathbb{C}_I}$ where $\mathbb{C}_I$ is Cohen ...
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161 views

The ethics of Borel determinacy

I was speaking with a friend the other day, and I happened to say "morally, Borel determinacy is as strong as ZF." I was riffing on the well-known result of Harvey Friedman, that we need ...
2
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1answer
43 views

Let $X\subset \mathbb{R}$ be Borel measurable. Can it be that $\aleph_0 <|X|<2^{\aleph_0}$?

I want to know if every Borel measurable set in the real line has cardinality either that of the naturals or of the reals. Of course the Continuum Hypothesis is not assumed. It is clear that every ...
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1answer
24 views

definition of countably generated Borel space

I have one source http://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0089999-2/S0002-9947-1957-0089999-2.pdf page 137, first paragraph of section 2 which says a countably generated Borel ...
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2answers
79 views

Dense-in-itself open sets in a subspace of the real line

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of ...
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0answers
37 views

$f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of measurable functions

Let $X$ be a measurable space and $Y$ a Polish space. I want to show the following equivalence. $f:X \to F(Y)$ is measurable iff $f^{-1}(\{\emptyset\})$ is measurable and there is a sequence of ...
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1answer
42 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
2
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1answer
47 views

Lusin space, isolated point

I have a question about Lusin space. Definition A Hausdorff topological space, $(X,\tau)$ is said to be a Lusin space if, there exists a topology $\tau'$ (on $X$) stronger than $\tau$ such that ...
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1answer
59 views

Informal interpretation of meager sets

I've been wondering if there is a nice informal interpretation of meager sets akin to the respective interpretations I give below to other notions of "small" sets. The general setup to tease out ...
2
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1answer
74 views

Pointclass of $\text{dom}(F)$ where $F:\omega^\omega\rightarrow\omega^\omega$ is partial recursive.

The definition I am working with: A partial function $F:\omega^\omega\rightarrow\omega^\omega$ is said to be partial recursive iff the partial function $G:\omega^\omega\times\omega\rightarrow\omega$ ...
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2answers
111 views

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
3
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1answer
44 views

Are measurable sets closed under projections?

For the following, let us assume that large enough sets to carry the arguments through do exist, i.e. that there are supercompact cardinals or whatever is sufficient. I know that all projective ...
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1answer
26 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
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1answer
36 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
5
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1answer
93 views

No Borel well-order of the reals?

I'm told there is no Borel well-order of the reals (in ZFC). I'm told, in fact, that this is because of Borel determinacy. However, this is usually a vague handwave of the form (a) take the usual ...
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1answer
24 views

$\{K∈K(X):K⊆U\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$

Let $X$ be a Polish space. The family of set $(i)$ $\{K∈K(X):K⊆U\}$ $(ii)$ $\{K∈K(X):K∩U≠∅\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$ where $K(X)$ is the space of all compact subsets of ...
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2answers
35 views

What's the difference between $ω^{<ω}$ and $ω^{ω}$?

I think that the first is all finite sequences of natural numbers, while the second is all infinite sequences of natural numbers. However I'm not sure how to draw the tree for $ω^{<ω}$, nor can I ...
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0answers
40 views

Are clopen sets Borel?

I understand that open sets, closed sets, $F_\sigma$-sets (countable unions of closed sets), $G_\delta$-sets (countable intersections of open sets), $F_{\sigma\delta}$-sets, $G_{\delta\sigma}$-sets ...
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1answer
28 views

Coanalytic families of Banach spaces

Is it true that if $G$ is a coanalytic family of separable Banach spaces, which is not Borel, and $H\subset G$ is not Borel, then H is coanalytic? This is something I have come across. I am reading a ...
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1answer
38 views

Borel image of a Polish space

Let $X, Y$ be Polish spaces and $f : Y \to X$ be a Borel function, i.e. preimage of every Borel subset of $X$ be a Borel subset of $Y$. Prove that $f[Y]$ is an analytic subset of $X$. Note: If ...
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1answer
21 views

Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. Where $\mathcal{C}=2^\mathbb{N}$.

If $X$ be a nonempty perfect Polish space, $Y$ a second countable space, and $f.X\to Y$ be inyective and Baire measurable. Question: Exist a homeomorphic copy of $\mathcal{C}$ contained in $f(X)$. ...
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50 views

$A=\bigcup_{i \in I}A_i$ is meager.

Let $X$ be a Polish space. Let $(I,<)$ be a wellordered set and $(A_i)_{i \in I}$ a family of sets meager in $X$. Let $A=\bigcup_{i \in I}A_i$. If consider the relation $x \leq^{*} y$ defined by: ...
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45 views

Show that there is a Borel function from [0,1] to $2^ω$

Show that there is a Borel function from [0,1] to $2^ω$. Conclude that there is a Borel function from $[0,1]^ω$ to $2^ω$. I'm not sure how to go about this. I know that a function is Borel if the ...
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42 views

Every perfect Polish space is a continuous 1-1 image of the Baire space

I am learning descriptive set theory by myself and stumble upon this theorem without proof, could somebody help me prove it: Every perfect Polish space is a continuous 1-1 image of the Baire space ...
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1answer
62 views

A $G_δ$ subset of $2^ω$ that is homeomorphic to $ω^ω$

How do I show that there is a $G_δ$ subset of the Cantor space $2^ω$ that is homeomorphic to the Baire space $ω^ω$? I've been given the hint to consider $G = \{x ∈ 2^ω : x\text{ is not eventually ...
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28 views

How many Borel sets are on an infinite Polish Space?

I am learning Descriptive set theory, and my question is how many Borel sets are on an infinite Polish space? I think they have the cardinality of the continuum, but I just do not see how to prove ...
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22 views

Understanding C-universal classes

There is a theorem in the book Classical Descriptive Set Theory, A. Kechris. In page 168 he states the following theorem: Let $X$ be a separable metrizable space. Then for each $\xi \geq 1$, there is ...
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27 views

$\text{BP}(X)/\text{MGR}(X)$ be the quotient space $\{[A]:A \in \text{BP}(X)\}$

Let $X$ be a topological space. $(a)$ $\text{BP}(X)$ denote σ-algebra of subsets of $X$ with the Baire Property $\text{BP}$. $(b)$ $\text{MGR}(X)$ denote the σ-ideal of meager sets in $X$. Let ...
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1answer
43 views

The inverse image of dense set is dense and of a comeager set is comeager?.

Let $X,Y$ be topological and $f:X \to Y$ be open and continuos. I am studying Baire space and I would like to try the following facts : $(i)$ The inverse image of dense set is dense. ...
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2answers
31 views

$X\times Y$ is second countable Baire spaces

If $X,Y$ are second countable Baire spaces, so is $X\times Y$. ? If true, a suggestion to try thanks
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30 views

measurable selection for almost-minimizers of an irregular functional

I'm faced with the following problem: I have a functional $F$ defined on $H^1$ curves $[0,1] \rightarrow \Omega \subset \mathbb{R}^n$ where $\Omega$ is either a compact subset or the whole ...
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2answers
61 views

Increasing sequence of open sets in a separable metric space.

Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...
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42 views

Baire space, analytic sets

I am reading a proof from the book 'Banach spaces and descriptive set theory, lemma 7.2' and it uses the following argument: If $A\subset X$ is analytic and $X$ is a closed subspace of the Baire space ...
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68 views

Borel sigma algebra not containing all subsets of $\mathbb{R}$?

Consider the smallest sigma algebra $\mathscr{B}$ generated by all open subsets of $\mathbb{R}$. One would expect that $\mathscr{B}$ contains all subsets of $\mathbb{R}$, but as it turns out, if we ...
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34 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
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1answer
50 views

Proof that the clopen subsets of $A^{\mathbb N}$ are finite unions using König's Lemma

Let $A$ denote a finite set, equippted with the discrete topology, and consider the product topology on $A^{\mathbb N}$, i.e. the set of all infinite sequences over $A$. Denote by $\pi_i : A^{\mathbb ...
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1answer
62 views

Covering/packing lemma for a Polish space with a Borel probability measure

Let $(X,d)$ be a complete separable metric space (Polish space) with a Borel probability measure $\mu$. Given $\varepsilon_1, \varepsilon_2 > 0$ can one find a finite set of disjoint open balls ...
4
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1answer
89 views

A conceptual link between trees and Polish spaces

Could somebody explain to me why trees are so relevant for the study of Polish spaces and descriptive set theory? I still do not get the proper connection (...and when I think I got it – see the ...
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3answers
65 views

Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a ...
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1answer
43 views

Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$ such that $\mathcal{U}\subseteq 2^\mathbb{N}$ is non-principal.

Let $\mathcal{U}$ now be an ultrafilter on $\mathbb{N}$. View $\mathcal{U}$ as a subset of $2^\mathbb{N}$. Question: If $\mathcal{U}$ is non-principal then $\mathcal{U}$ does not have the Baire ...
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1answer
48 views

Question of regular open

A set $U$ in a topological space $X$ is called regular open if $U=\text{Int}\left(\overline{U}\right)$. Similarly, a set $F$ is regular closed if $X\setminus F$ is regular open or equivalently ...
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1answer
34 views

Question about the Baire space, $\sigma$-algebra and $\sigma$-ideal.

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$. Assume $X$ is second countable Baire ...