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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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
88 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
32 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
22 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
29 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
85 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
19 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
31 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
32 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 ...
0
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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
29 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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Illfounded trees as “retract” of all trees [migrated]

Definitions: Let $\omega^{<\omega}$ be the set of all finite sequences of natural numbers. For $u, v \in \omega^{<\omega}$, let $u \prec v$ denote that $u$ is a prefix of $v$. We call a subset ...
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1answer
18 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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45 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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0answers
44 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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0answers
25 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 ...
2
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1answer
43 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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0answers
25 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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0answers
20 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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0answers
26 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
22 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
29 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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21 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
56 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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0answers
26 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 ...
4
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0answers
57 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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0answers
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
40 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 ...
3
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1answer
55 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 ...
4
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3answers
64 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 ...
2
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1answer
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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 ...
0
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1answer
44 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 ...
3
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1answer
28 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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1answer
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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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0answers
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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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0answers
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 ...
3
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1answer
62 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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2answers
28 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
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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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0answers
26 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
30 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
35 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 ...
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1answer
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Question about of countable dense subsets of spaces perfects Polish [closed]

Let $X$ be a perfect Polish space. If $Q \subseteq X$ be countable dense. It is has $Q$ is $F_{\sigma}$ but not $G_{\delta}$. ? Any ideas. thanks.
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1answer
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The collection of all compact finite subsets is $F_{\sigma}$ in the hyperspace of all compact subsets.

Let $X$ be metrizable (not necessarily Polish), and consider the hyperspace of all compact subsets of $X$, $K(X)$, endowed with the Vietoris topology (subbasic opens: $\{K∈K(X):K\subseteq U \}$ and ...
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1answer
33 views

Open neighbourhoods, Polish spaces, and basis for the Baire space

This is a follow-up of a question I asked yesterday answered by GEdgar. I think I see now GEdgar’s answer, but I am not sure about an issue related to it. Thus, I will write my general understanding ...
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2answers
47 views

Open neighbourhoods & polish spaces - typo in Marker's notes?

A very (very!) easy question that merges together the very basic concepts of two fields that I find always problematic for my understanding, namely topology and descriptive set theory. Everything ...
3
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1answer
40 views

Existence of schemes Souslin in separable spaces.

If $X$ is nonempty separable we can build a Souslin scheme $U_{s}$ such that $U_{\emptyset}=X$, $U_{s}$ open nonempty, $\overline{U}_{s\hat {}i}\subseteq U_{s}$, $U=\bigcup_{i}U_{s\hat {}i}$ and ...
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1answer
45 views

Initial segments of trees

I am fairly sure this question will look quite trivial, but I have some trouble getting the reason behind the use of the symbole $\subset$ when we deal with sequences and initial segments of ...
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1answer
41 views

This question is about the scheme Souslin

A Souslin scheme on a set $X$ is family $(A_{s})_{s \in{\omega^{<\omega} }}$ of subsets of $X$. If $(X,d)$ is a metric space, we say again that $(A_s)$ has vanishing diameter if diam$(A_{x| n })$ ...