# 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 ...
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 ...
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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 $\mathbb N^{\mathbb N}$ ...
1 vote
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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 ...
1 vote
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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. ...
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 ...
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 $X\setminus A$ ...
1 vote
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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 ...
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 ...
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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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139 views

1 vote