# 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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1 vote
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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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1 vote
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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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### 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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### 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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### Every compact subset of Baire Space (ω^ω) has empty interior [duplicate]

Here is the Well known proposition. I assumed that K is the compact subset of ω^ω and it contains an open set Ns, so I thought if K is compact so Ns is because Ns is clopen set, If I can not find a ...
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### Every compact subset of Baire space $\omega^\omega$ has empty interior. [duplicate]

Here is the well-known proposition, I assumed that $K$ is compact subset of $\omega^\omega$ and it contains an open Set $N_s$, so $N_s$ is clopen and thus it is compact. If I can show $N_s$ can not be ...
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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 ...
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