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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Complement of any dense countable subset of reals is homeomorphic to irrationals
I recently stumbled upon this:
For any infinite countable subset $A\subseteq\mathbb R$ such that $\overline A=\mathbb R$, the complement $\mathbb R\setminus A$ is homeomorphic to the Baire space. (Or,...
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Variant of Baire theorem
I consider $(X,d)$ a complete metric space. I have this weak form of the Baire theorem : There does not exist nonempty open subset $O$ of $X$ such that $O=\bigcup_{n\geq 0} F_n$ where the $F_n$ are ...
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Is this set countable for any function?
Working with CH I order ($\prec$) $^\omega\omega$ of order type $\omega_1$. I let $$f_\alpha:=\min_\prec\{f\in\hspace{1mm}^\omega\omega:\neg (f(n) \leq f_\beta(n)),\forall n\in\omega,\beta\in\alpha\}$$...
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Can you construct a sequence to witness that this set isn't compact? [closed]
I know that if $K$ is a closed subset of $^\omega\omega=:\mathcal{N}$ and there exists an $f\in\mathcal{N}$ such that $K\subseteq \{g\in \mathcal{N}:g\leq f \}$ (where $g\leq f$ is pointwise), then $K$...
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How is the set $C(f)\cap V$ of second category in $V$?
I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the ...
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Must a subset of the real line that is comprised entirely of condensation points be a Baire space?
Let $X$ be a subset of the real line in which every point is a condensation point. Is $X$ a Baire space?
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Is this proof that the irrational number set is a Baire space correct?
I know there's proof that the irrational number set is a Baire space, but when I tried to prove this, I used a different way. However, I feel my proof is not right.
Let $I= \mathbb{R} \backslash \...
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Nowhere dense subsets of a dense subspace are nowhere dense in the whole space and vice versa
It seems, with the following lemma, the proposition at the bottom easily follows.
If $Y\subset X$ dense. Then, for every nonempty $A\subset Y$, $\text{Int}_Y \left(\text{cl}_Y A\right)=Y\cap \text{...
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Equivalent definitions of Baire spaces
We say that a metric space $X$ is a Baire space if there is no open set $E$ such that $$E \subseteq \bigcup\limits_{n\geq 1} F_i,$$ in which each $F_i$ is a closed set with empty interior.
Suppose ...
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Excercise 15 Rudin functional analysis chapter 2
I am self-studying the book function analysis of Rudin. I got stuck on the final passage of the following exercise.
Suppose $X$ is an $F-$space (a topological vector space with a topology induced by a ...
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Examples of dense and codense $G_\delta$ subsets of $\mathbb{R}^2$ that are not homeomorphic to $\mathcal{N}=\mathbb{N}^\mathbb{N}$
I have been asked to show that if $X$ is a dense $G_\delta$ subset of $\mathbb{R}$ such that $\mathbb{R}\setminus X$ is also dense in $\mathbb{R}$, then $X$ is homeomorphic to $\mathcal{N}=\mathbb{N}^\...
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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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Does there exist a bijective, continuous map from the irrationals onto the reals?
Let $\mathbb{P}$ be the irrational numbers as a subspace of the real numbers. $\mathbb{P}$ is homeomorphic to $\mathbb{N}^\mathbb{N}$, which is also called the Baire space.
It is well known, and ...
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