# Tagged Questions

This tag is intended for questions on topics related to Baire category, such as Baire category theorem, meager sets (set of first category), nonmeager sets (set of second category), Baire spaces etc.

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### Show that C^α([0, 1]) is of first category in C([0, 1]).

Recall that for any $0 < α < 1,$ the space $C^\alpha ([0, 1])$ is the set of continuous functions on $[0, 1]$ with norm of f = sup |f| + $sup \ x\ne y$ |f(x) − f(y)|/|x − y|^α< ∞, equipped ...
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### Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
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### Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense.

Let $f: \mathbb R \rightarrow \mathbb R$ be a continuous function that is nonconstant on any interval. Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense. My ...
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### Why need open in the Baire Category Theorem

In the statement of the Baire Category Theorem, one needs to include openness of a set so that the theorem holds. Question: what is the example such that countable intersection of dense sets is not ...
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### Baire category of subset $\mathbb{R} \setminus B$ with lebesgue zero measure $B$

Let $\mathbb{Q}=\{ q_i\}^{\infty}_{i=1}$. Let $I_{ij} = \left(q_i - \delta_{ij}, q_i + \delta_{ij}\right)$, $\delta_{ij}=2^{-i-j-1}$ for all $i,j \geq 1$ be an open interval. For every $j\geq1$, ...
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### Baire's Theorem and Irrationals

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
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### Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces don'...
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### If $f:\mathbb R \to \mathbb R$ is an additive function whose graph is $G_{\delta}$ in $\mathbb R^2$ , then the graph is closed in $\mathbb R^2$?

If $f:\mathbb R \to \mathbb R$ is an additive function i.e. $f(x+y)=f(x)+f(y) ,\forall x,y \in \mathbb R$ such that $G(f):\{(x,f(x)) : x\in \mathbb R\}$ is a countable intersection of open sets , ...
Let $X$ be a metric space. Are these conditions equivalent: Each set of the 1. category in $X$ has empty interior; $X$ is of the 2. category. It is obvious that $1 \Rightarrow 2$. Is it true that $... 1answer 42 views ### Why empty set? (consequences of Baire's theorem) I did not understand the proof of Theorem 5.13 of Rudin, [Real and Complex Analysis]. See next. In a complete metric space X which has no isolated points, no countable dense set is a$G_{\delta}$. ... 1answer 46 views ### Question about the Baire space,$\sigma$-algebra and$\sigma$-ideal. Let$\text{BP}(X)$denote$\sigma$-algebra of subsets of$X$with the Baire Property BP and$\text{MGR}(X)$denote the$\sigma$-ideal of meager sets in$X$. Assume$X$is second countable Baire space.... 1answer 72 views ### Question about of Baire property and Baire space In reading Kechris book. Please, I would like help with this proposition. For convencion we put for$A \subseteq X$, $$\sim A=X\setminus A$$ If$A$is comeager in$U$, we say that$U$forces$A$, ... 0answers 30 views ### Proving a linear function is bounded using the Baire category theorem (or its consequences) This is a problem from Folland. Let$\mathcal{X}, \mathcal{Y}$be Banach spaces. If$T : \mathcal{X} \rightarrow \mathcal{Y}$is linear and$f \circ T \in \mathcal{X}^*$for all$f \in \mathcal{Y}^*,...
Let $(X,T)$ be a topological space. When we say $A$ is first category in $(X,T)$ we mean that it is the union of (or is covered by) countably many sets which are nowhere dense in $(X,T)$. $A$ is ...