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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Is there an indecomposable topological group structure on the Baire Space?

Follow up to this follow up question. Is there an indecomposable group $G$ ie, a group that can't be written in the form $A\times B$ where both $A$ and $B$ are nontrivial groups along with a topology ...
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Are all topological group structures on the Baire Space of the form $G^{\mathbb N}$ for $G$ a countable discrete group?

This question was accidentally trivial: For any countably infinite discrete group $G$ we have that $G^{\mathbb N}$ is a topological group structure on the Baire Space, as pointed in Qiaochu Yuan's ...
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Is there a topological group structure on the Baire Space?

The sum of two irrationals might not be irrational so we can't use that; Also $\mathbb N$ is not a group so there's no obvious way to define a group structure in it seen as $\mathbb N^{\mathbb N}$ ...
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Non empty perfect set and it's cardinality in different spaces

Let $(X, \tau) $ be any topological space. $P\subset X$ is called perfect if $P'=P$ where $P'$ is the set of all limit ponits of $P$. If $(X, \tau) $ is a $T_1$ space, then any open set containing a ...
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Homeomorphism between $\mathbb N^\infty$ and a closed subset $\mathbb M$ of $(\mathbb N^\infty)^\infty$

Trying to figure out a proof of a lemma that I'm reading in Stochastic Relations by Ernst-Erich Doberkat. The Baire space, denoted $\mathbb{N}^\infty$, is the infinite product of the natural numbers. ...
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Example of function of Baire 3 and Baire 4

i'm looking for explicit examples of real-valued functions of the Baire third and fourth class, without using Borel-measurability but just using some characterization theorem of the previous classes. (...
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Why Conway 13 base function is Baire two?

I'm writing my thesis about Baire Classes. I want to prove that the 13 base Conway function is in the Baire two class. I need a clear proof of this fact, a proof in which is described how to "...
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Baire space and increasing union of closed subspaces

Let $X$ be a Baire space. Suppose there is an increasing sequence $C_1\subset C_2\subset \cdots $ of closed subspaces of $X$, whose set-theoretical union is $X$. Since $X$ is Baire, we know that some ...
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Can we classify all topological space $(X, \tau) $ where every second category sets are Residual sets?

$(X, \tau) $ be a topological space. $A\subset X$ is Residual if $X\setminus A$ is of first category. In a Baire space, a Residual set is of second category. $A\subset X$ Residual, then $X\setminus A$ ...
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Baire space is homeomorphic to countably many copies of itself

On wikipedia I found that the Baire space $\mathcal{N}$ is homeomorphic to the product of a countable number of copies of itself, however, I haven't been able to find a proof. The Baire space is ...
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Is $\mathbb{Q} \times \mathbb{Q}$ a $G_\delta$ set?

I can prove that $\mathbb{Q}$ is not a $G_\delta$ set in $\mathbb{R}$. I was applying the same Baire space argument to show that $\mathbb{Q} \times \mathbb{Q}$ is not a $G_\delta$ set. I was thinking ...
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The Baire space is homogeneous,

it's kwown that the Cantor space $2^{\omega}$ is a homogeneous topological space (because it is a topological group). Does anyone have any idea why $\omega^{\omega}$ is a homogeneous topological space?...
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$\bigcap_{n\in\mathbb{N}}{F_{n}}$ is dense in $X_{0}$

Let $\{(X_{n},d_{n})\}_{n\in\mathbb{N}}$ be a sequence of complete metric spaces. If $\{f_{n}\colon X_{n}\to X_{n-1}\}_{n\in\mathbb{N}}$ is a sequence of functions continuous such that \begin{equation}...
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$X$, $Y$ be metric spaces and $f: X \to Y$. If $X$ is Baire and $Y$ is separable then $f$ is continuous in a dense $G_{\delta}$ of $X$

Let $X$, $Y$ be metric spaces and $f: X \to Y$. Suppose $X$ is Baire and $Y$ is separable. If $f ^{− 1} (O)$ is $F_{\sigma}$ for every open $O \subset Y$, show that $f$ is continuous in a dense $G_{\...
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If $f_{n}$ has a dense image, then $\bigcap (f_{1}\circ\cdots\circ f_{n})(X_{n})$ is dense

Let $\{(X_{n}, d_{n})\}_{n\in\mathbb{N}}$ be a sequence of complete metric spaces and $\{f_{n}: X_{n}\to X_{n − 1}\}_{n\in \mathbb{N}}$ a sequence of continuous functions. If $f_{n}$ has a dense image ...
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Description of a set in the Baire space.

I have the following exercise. In the Baire space, $\omega ^ \omega$ ,describe a set $A$ such that $int(A)\neq \emptyset$ and $A\neq \bar{A}$ I recently just got to know the Baire space, it is ...
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Difficulty showing a dense $G_{\delta}$ subset of a Baire space is Baire

I have been working on showing that the irrationals is a Baire space. So far I have shown that the irrationals can be expressed as a $G_{\delta}$ set and I know that if this set was to be Baire then ...
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A $G_{\delta}$ subset of a Baire space is Baire

I have been seeing this fact used a lot but have not been able to find a proper proof justifying it. Would anyone be able to outline one?
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On Čech-complete space.

I'm reading an article of topology and i came across a Properties : Properties : Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete Every Čech-complete space is a ...
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Recursive representation of closed sets in Baire Space ($\Sigma^1_1$)

These notes (pg 2) say that A set $C \subseteq \omega^\omega$ is closed if and only if there is an $z \in \omega^\omega$ and a recursive predicate $R \subseteq \omega^{<\omega}\times\omega^{<\...
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Equivalent formulations of analytic/ $\omega$-suslin sets (in Baire Space) [duplicate]

I'm aware that there are more general formulations of these concepts, but I'm just starting to learn these and have been looking at them in a very restricted and simple context. Let $A \subset \omega^{...
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Does there exist a second category set that is not a Baire space?

Can someone give an example of a second-category set $Y$ in a metric space $X$ but $Y$ is not a Baire space. We know that Baire space$\implies $ Second Category. But I am trying to show that the ...
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Clarification of Baire category theorem: a (counter-?)example

I am trying to understand the statement of Baire's category theorem. Why is $\bigcap_{n \in \mathbb{N}} A_n := \bigcap_{n \in \mathbb{N}} \big[(n, n+\frac{1}{2}) \cap \mathbb{Q}\big] = \emptyset$ ...
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Wadge Hierarchy on $\Delta^{0}_{2}$ sets

i'm studying the Wadge hierarchy on Baire space and Cantor space. I'm asking whether or not the $\Delta^0_2$ sets form a unique degree in these spaces and why the $\Sigma^0_2$-true (i.e. not polish) ...
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Showing $A\cap B \ne \emptyset$

So I've been trying to prove this problem for the past week but I don't know if the path chosen was prudent. I used Baire's Theorem to arrive at my proof for the problem. May someone kindly provide ...
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Operation over the Baire space $\omega^\omega$ that preserves Borel sets

We have the following operation on sequences of natural numbers (elements of $\omega^\omega$): $$\begin{align}*:\omega^\omega\times\omega^\omega &\longrightarrow \omega^\omega\\ (x,y) &\...
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Continuous parametrization of continuous functions in the Baire space $\omega^\omega$

It can be shown that in the Baire space there is a bijection $$\begin{align}\mathcal{N} &\longrightarrow \{f \in {}^\mathcal{N}\mathcal{N} \mid f \text{ is Lipschitz}\}\\ x &\longmapsto \ell_x\...
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A subset of the second Baire category on the real line

Why is the subset in $\mathbb{R}$ of the second Baire category uncountable?
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A question regarding the convergence of a sequence of functions at some point on $[0,1]$

Problem: Does there exist a sequence of continuous functions $ f_n:[0,1] \to [0,\infty)$ such that $\lim_{n \to \infty} \int_0^1 f_n(x) dx=0$ but their doesn't exist any $x \in [0,1]$ for which the ...
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$\Bbb R^J$ is a Baire space in product topology

I have written the following proof of the fact that, $\Bbb R^J$ is Baire space in product topology. Can anyone check my proof and say about any fault? Thanks in advance. $\textbf{Proof :---}$ ...
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Q(set of rationals) is of baire's first category in itself but N(natural numbers) are of second category in itself .

since we need to see whether they can be written as a countable union of nowhere dense sets or not . for N , i thought {1} these single-tons are dense in N . so N is of second category. Is this ...
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Baire space has isolated point

Let $X$ be a Hausdorff, Baire space. I want to prove that $X$ has an isolated point. In a Hausdorff space, singletons $\{x\}$ are closed. Now suppose for a contradiction $X$ has no isolated points. ...
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Convergence in the Baire space $\mathcal{N}=\mathbb{N}^{\mathbb{N}}$

$\mathcal{N}:=\mathbb{N}^{\mathbb{N}}$ Let $d(m,n)=\sum\limits_{n=1}^{\infty}\:\frac{1}{2^n}\: d_n(m_n,n_n)$ be the product metric on $\mathcal{N}$, where $d_n$ denotes the discrete metric. I want to ...
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How to show that $\mathbb{N}$ is a Baire space [duplicate]

I read that a topological space $(X,d)$ is a Baire space if for every sequence $\{X_n\}$ of open dense subsets of $X$, the set $\bigcap_{n=1}^{\infty}X_n$ is also dense in $X$. Since every complete ...
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Polish spaces are continuous images of the Baire space

I'm having some troubles understanding the proof of Theorem 7.9 (pag. 39) in Kechris' "Classical Descriptive Set Theory": There are two points of the proof proposed that I don't quite understand. ...
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Let $X$ be a complete metric space. $E$ is a non empty open set in $X$ Then is $E$ a first category set or second category set.

Let $X$ be a complete metric space. $E$ is a non-empty open subset of $X$ Then is $E$ a first category set or a second category set. it seems to me that $E$ should be a second category set, but how ...
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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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9 votes
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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,$\mathbb{Q}=...
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