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Questions tagged [the-baire-space]

For questions about the Baire space, that is the family $\mathbb{N}^\mathbb{N}$ of all sequences of natural numbers with the product topology. For questions about the class of Baire spaces - spaces in which Baire category theorem holds - use (baire-category) tag.

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$H$ thin in $X$ implies that $X\setminus H$ contains a generic subset

I am studying Baire Theorem on Functional Analysis and I am dealing with the following definitions (translated from portuguese): A subset of a topological space $ X $ is said to be rare in $ X $ if ...
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What can I say about $A^{c}$ if $A$ is not open

Let $X$ be some topological space: I want to prove that $i)\Rightarrow ii)$ where: $i)$ Let $A \neq \varnothing$ and $ A\subset X$ open, then $A$ is non-meagre. $ii)$ Let $A\subset X$ be meagre, ...
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Baire space $\mathbb{N}^\mathbb{N}$ written as $\mathbb{R}$ [duplicate]

I'm writing my bachelor thesis on various topics from set theory and descriptive set theory (mainly topological games), and I just read a paper in which the Baire Space $\mathbb{N}^\mathbb{N}$ is ...
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$S_\infty$ is a non-locally compact Polish group (Kechris)

Here is example $7)$, pp. $59$ of Kechris' book "Classical Descriptive Set Theory": Let $S_\infty$ be the group of permutations of $\mathbb{N}$. With the relative topology as a subset of $\...
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Two Baire measure, $\left\{A\subseteq X: A \text{ Baire set}\ \mu(A)=\nu(A)\right\}$ is $\sigma$-algebra

pag 235 Barry Simon. A comprehensirve course in analysis Let $X$ compact hausdorff space. Let $\mu,\nu$ baire measures, and $\mu(X)=\nu(X)=1.$ Let $S:=\left\{A\subset X: A \text{ baire set and } \mu(...
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The Baire space as an automorphism group?

Wikipedia claims that the Baire space "is [the] automorphism group of [a] countably infinite saturated model $\mathfrak{M}$ of some complete theory $T$", however I see neither an obvious group ...
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Baire Space and Its Equivalent Property

Lemma $X$ is a Baire space if and only if given any countable collection $\{U_n\}$ of open sets in $X$, each of which is dense in $X$, their intersection $\cap U_n$ is also dense in X. A space X is ...
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Gale-Stewart game

Let's look at the Gale-Stewart game on $\mathbb N^{\mathbb N}$ space (it consists of infinite sequences of natural numbers). There are 2 players: $A$ and $B$. They write one natural number by turns (A ...
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A Homeomorphism Between the Baire Space and a Countable $G_\delta$ set of the Cantor Space

I came across the following fact in Alex. Kechris' Classical Descriptive Set Theory (Ex. 3.12, pg. 17, 1994): Let $0^n$ be a string of $n$ $0$s. Then the map $f(x)=0^{x_0}10^{x_1}10^{x_2}\cdots$, ...
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On the complexity of some $\sigma$-ideals in the Baire space

Let $I$ be a Borel generated $\sigma$-ideal on the Baire space. We say that this ideal is $\Sigma^1_2$ if $$\{c \in \omega^\omega\ |\ c\text{ is a Borel code and }B_c \in I\} \in \Sigma^1_2.$$ Where $...
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Gale-Stewart Theorem (open games are determined) implies closed games are determined

A Gale-Stewart game $G(A)$ is played on a set $A\subseteq\mathbb N^\mathbb N$. In this game, players p0 and p1 alternately pick a natural number, forming a sequence $\alpha:=\alpha_0\alpha_1\alpha_2\...
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Is zero-dimensional, separable metric space a $\sigma$-space?

Let $X$ be a zero-dimensional, separable metric space. Is then $X$ a $\sigma$-space (i.e. every $G_\delta$ subset is $F_\sigma$ in $X$)? I do not know.
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Is every boldface $\mathbf{\Pi^0_2}$-definable quotient of Baire space Polish?

I think it's true that every Polish space is a boldface $\mathbf{\Pi^0_2}$-definable quotient of Baire space. However, is the converse true? That is, is every $\mathbf{\Pi^0_2}$-definable quotient of ...
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The rationals are an intersection of open set-contradiction to Baire theorem

Prove or contradict: There are infinite open sets $U_1,U_2,...\in \mathbb{R}$ such that : $\mathbb{Q}=\bigcap^\infty _{i=1} U_i $ So I saw the following answer: No, becuase if it was true,$\...
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Application of Baire theorem in complete metric space [closed]

This is not a homework assignment, its a question from a topology test which im trying to solve to prepare for my own. I dont know how to solve it, so anything helpful will be appreciated. Let $(X,...
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Reference request: lattice theory of the algebra of open sets in a subspace of the Baire space

I'm interested in lattice-theoretic properties of the algebra of open sets in a subspace of the Baire space $\Bbb N^{\Bbb N}$ can or cannot have. What is a good source on this matter? (An Internet ...
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Is there a perfect Polish topology on the Baire space that is strictly finer than the usual topology?

The question is: is there a perfect Polish topology on the Baire space $\mathbb{N} ^ \mathbb{N}$ that is strictly finer than the usual topology on $\mathbb{N} ^ \mathbb{N}$? The usual topology is the ...
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$T$ linear. $T$ is bounded if, and only if, $int(T^{-1}(\overline{B_{N_2}}(0,1))) \neq \emptyset$R

$T: N_1 \rightarrow N_2$ linear. Let $A = \overline{B_{N_2}}(0,1)$. As $T$ is bounded, $T^{-1}(A)$ is a closed set in $N_1$, so $T^{-1}(A)$ is complete. Moreover, $T^{-1}(A)$ is a Baire space, since ...
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Something about the Baire Space

I do not know why we put forward the "Baire Space"? What is the difference between the Baire Space and Metric Space? Can you give me some examples? Thank you very much!
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Is every Polish space the quotient of some subset of Baire space?

Question is as in the title: Is every Polish space the quotient space of some subset of Baire space $\omega^\omega$ ?
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Base of the Baire space [closed]

Why base of the Baire space is countable? Because it is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points.
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Are the basic open sets of the Baire space closed?

One way to describe the topology of the Baire space $\mathbb{B} = \omega^\omega$ is that the basic open sets are of the form $N_\eta = \left\{ f \in \omega^\omega \middle |\ \eta \subseteq f \right\}$...
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Any $G \subseteq \mathbb{N}^\mathbb{N}$ is clopen

I have a question concerning the space $\mathbb{N}^\mathbb{N}$. I found in Srivastava's book on Borel sets that the sets of the form $$\Sigma (s) := \{ \alpha \in \mathbb{N}^\mathbb{N} \ | \ s \prec ...
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1answer
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Confusion About Pointclasses

I am doing some work learning about the Axiom of Determinacy and its consequences. This has led me to learning about the properties of the Baire space, $\omega^\omega$. I have recently come across the ...
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If a ball is the union of two sets, does one of them non empty interior?

Hi everyone: Suppose that an open ball $B$ of $\mathbb{R}^{m}$ $(m\geq2)$ can be written as the disjoint union of two sets: $B=E\cup F$. Can we conclude that one these sets,$E$ or $F$, has non empty ...
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Prove that the Set $P$ of algebraic polynomial is a first category set in $C[a,b]$

Prove that the set $P$ of algebraic polynomial is a first category set in $C[a, b]$ I know the definition of first category is countable union of nowhere dense sets. and further more I know that the ...
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Why the set $\{f \in \omega^\omega : \exists_{m}\forall_{m \leq n}^{}(f(n)\neq g(n))\}$ is $F_{\sigma}$ and meager?

For any $g \in \omega^\omega$, Why the set $\{f \in \omega^\omega : \exists_{m}\forall_{m \leq n}^{}(f(n)\neq g(n))\}$ is $F_{\sigma}$ and meager?. I do not know why this set is $F_{\sigma}$ and ...
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The existence of a type of dense subsequence of every dense sequence of eventually zero functions in $\omega^\omega$

Let $G=\{g_n: n \in \omega\}\subseteq \omega^\omega $ where $g_n$ is eventually equal to zero. Clearly $G$ is dense in $\omega^\omega$. Suppose that $Q=\{q_n:n \in \omega\}\subseteq G$ is also dense. ...
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Complete metric space of sequence of positive integers [duplicate]

Let $(A,d)$ be the space $\mathbb{N}^{\mathbb{N}}$ of sequences of positive integers where $d((a_i)_i, (b_i)_i)= \frac{1}{n}$ where $n$ is the least coordinate at which $(x_i)_i$ and $(y_i)_i$ ...
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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 $\...
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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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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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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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How to prove continuity in Baire space?

Let $X=(\omega^\omega,d)$ be Baire space with the metric $d$ defined in assignment $1$. Define a function $G:X\to X$ by letting, for $f\in X$, the function $G(f)$ be defined by: $$(G(f))(n)=\begin{...
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Proving a metric space $\mathbb N^{\mathbb N}$ with $d(x,y)=1/\min\{j:x_j\neq y_j\}$ is complete

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= \...
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Is the Baire space $\sigma$-compact?

Is the Baire space $\sigma$-compact? The Baire space is the set $\mathbb{N}^\mathbb{N}$ of all sequences of natural numbers under the product topology taking $\mathbb{N}$ to be discrete. It is a ...
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Is the space $\mathbb N^ \mathbb N$ metrisable? [duplicate]

Given the space $\mathbb N^ \mathbb N$ with the topology generated by basis sets of the form: $$[V,n] = \{x \in \mathbb N^ \mathbb N ; V \text{ is an n prefix of x}\}$$ I can see that this space is ...
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are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent?

are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent? By the product topology on $\omega^\omega$ I mean the topology in which an open basis set is a set of ...
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$\mathcal{N}^\omega$ is homeomorphic to $\mathcal{N}$, where $\mathcal{N}$ is Baire space.

I am trying to prove that $\mathcal{N}^\omega$ is homeomorphic to $\mathcal{N}$ where $\mathcal{N}$ is Baire Space $\omega^\omega$, of all sequences of natural numbers, $\langle a_n;n \in \mathbb{N}\...
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What is the cardinality of $\omega^\omega$?

I have started reading about Baire Space as a prerequisity for the subject of Borel hierarchies. And if I understand correctly, Baire space is $\omega^\omega$ (The space of all functions from $\omega$ ...
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Prove that Baire space $\omega^\omega$ is completely metrizable?

When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the ...
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What can we say about closed sets in the Baire space that are neither open nor compact?

I'm trying to figure out what closed subsets in $\omega^{\omega}$ equipped with product topology should look like. It seems to me it's relatively easy to have an idea about compact closed subsets and ...
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Baire space homeomorphic to irrationals

I try to show that the Baire space $\Bbb N^{\Bbb N}$, with regular product metric, is homeomorphic to the unit interval of irrationals $(0,1)\setminus\Bbb Q$. I already know that the needed function ...
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Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.
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Proof that $\omega^\omega$ is completely metrizable and second countable

I have almost solved the following problem but am stuck at the very end, can you help me finish it? Thank you for your help. Let $n<\omega$ and $t\in {}^n\omega$. We define $U_t=\{s\in {}^\omega\...
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Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
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Basic questions about $\mathbb{Z}^{\mathbb{N}}$ with the product topology

can someone please let me know if the following is correct: 1) Let $\mathbb{Z}$ be the integers endowed with the discrete topology and $\mathbb{N}$ the natural numbers. Is $\mathbb{Z}^{\mathbb{N}}$ a ...
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$\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space

In my attempts to write a cleaner proof (compared to what I was taught) to the fact that for uncountable Polish spaces the Borel hierarchy does not stop until $\omega_1$ I am trying to prove the ...