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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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 ...
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38 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
17 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
75 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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32 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
41 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 ...
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1answer
34 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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39 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 $G:\omega^\omega\times\omega\rightarrow\omega$ ...
4
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2answers
107 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 ...
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1answer
38 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
35 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 ...
4
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1answer
88 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
22 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
32 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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36 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
32 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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19 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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48 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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31 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
58 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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26 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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21 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
34 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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26 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
58 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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37 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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63 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
47 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
58 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
41 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
45 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
33 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 ...
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59 views

Question about of Baire property and Baire space

In reading Kechris book. Please, I would like help with this proposition. For convencion we put for $A \subseteq X$, $$\sim A=X\setminus A$$ If $A$ is comeager in $U$, we say that $U$ forces $A$, ...
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23 views

Simple question about of the formulas concerning the forcing relation $U \Vdash A$

If $A$ is comeager in $U$, we say that $U$ forces $A$, in symbols $$U \Vdash A$$ A weak basis for a topological space $X$ is a collection of nonempty open sets such that every nonempty open set ...
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83 views

Identify subsets of $\mathbb{N}$ with their characteristic functions

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...
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1answer
64 views

Is there a subset $A\subseteq \mathbb{R}$, not having the BP (Baire property)?

Let $\mathcal{I}$ be the $\sigma$-ideal of meager sets of a topological space $X$. We say, that a set $A\subseteq X$ has the Baire property (BP) iff $A=^*U$ for some set $U\subseteq X$. Here $A=^*B$ ...
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3answers
34 views

The Hyperspace of Compact Sets is Hausdorff?

If $X$ is Hausdorff then $K(X)$ is Hausdorff where $K(X)$ is the Hyperspace of Compact Sets equipped with the topology from the Hausdorff metric.(subbasic opens: $\{K\in K(X):K\subseteq U\}$ and ...
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1answer
32 views

Exercise of comager set and the space $C([0,1])$

Let $X$ be a topological space and $P \subseteq X$ holds generic. If $P$ is comeager , we say that $P$ hold generically or that the generic element of $X$ is in $P$ Show that the generic element of ...
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28 views

Easy question about of space Polish

Let $X, Y$ be Polish spaces and $f:X\to Y$ continuous. Show that if $f(X)$ is uncountable, then there is a homeomorphic copy $K \subseteq X$ of $\mathcal{C}$ such that $f\upharpoonright K$ is ...
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1answer
34 views

The Hyperspace of Compact Sets is zero -dimensional

Let $K(X)$ the space of all non-empty compact subsets of $X$ equipped with the topology from the Hausdorff metric.(subbasic opens: $\{K\in K(X):K\subseteq U\}$ and $\{K\in K(X):K\cap U\neq ...
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2answers
36 views

Homeomorphic and space Polish

Let $X \subseteq \Re$ be $G_{\delta}$ and such that $X$, $\Re \setminus X$ are dense in $ \Re$. Then $X$ is homeomeorphic to $\mathcal{N}$. Also when replacing $ \Re$ by a zero-dimensional nonempty ...