# 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.

48 questions
1answer
21 views

### $H$ 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 ...
1answer
51 views

### 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, ...
0answers
98 views

### 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 ...
1answer
74 views

1answer
104 views

### 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 ...
1answer
17 views

### 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 ...
1answer
109 views

### 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 ...
2answers
100 views

### 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 $f(x)=0^{x_0}10^{x_1}10^{x_2}\cdots$, ...
1answer
137 views

2answers
76 views

### 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.
1answer
36 views

### 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 ...
2answers
160 views

0answers
84 views

### 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 ...
1answer
72 views

### 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 ...
1answer
35 views

### $T$ 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 ...
1answer
72 views

### 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!
2answers
130 views

### 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 $\omega^\omega$ ?
1answer
64 views

### 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.
1answer
72 views

### 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 $N_\eta = \left\{ f \in \omega^\omega \middle |\ \eta \subseteq f \right\}$...
1answer
70 views

1answer
396 views

1answer
76 views

### 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 ...
0answers
188 views

2answers
447 views

### 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 ...
1answer
346 views

### 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 ...
1answer
508 views

### $\Sigma_{\alpha+1}^0(X)$ 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 ...