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.

83 questions
Filter by
Sorted by
Tagged with
24 views

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 ...
23 views

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 ...
33 views

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 $\mathbb N^{\mathbb N}$ ...
1 vote
74 views

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 ...
• 8,938
1 vote
21 views

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. ...
53 views

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. (...
64 views

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 "...
90 views

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 ...
• 2,124
52 views

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 $X\setminus A$ ...
• 8,938
1 vote
80 views

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 ...
• 105
55 views

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 ...
• 95
46 views

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?...
• 35
65 views

$\bigcap_{n\in\mathbb{N}}{F_{n}}$ 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 ...
• 419
159 views

139 views

• 1,245
1 vote
67 views

• 703