# 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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### Baire related problem

Let $f:\Bbb R\to \Bbb R$, $f$ in $C^{\infty}$. Suppose that for all $x \in \Bbb R$, there exists an integer $n$ (which depends on $x$) such that $f^{n}(x) = 0$ ($f^{n}$ is composing $f$ $n$ times) ...
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### Baire space but not locally compact

I need an example that is a Baire space but not locally compact. I think, $\mathbb{R}^ \mathbb{R}$ is such an example. $\mathbb{R}^ \mathbb{R}$ is not locally compact. But I could not proof that it ...
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### Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense $G_{\delta}$...
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### $G_\delta$ set of nowhere differentiable functions?

I'm trying to show that in $C([0,1])$ with the supremum metric, there exists a dense $G_\delta$ set of nowhere differentiable functions. Honestly, I don't know how to approach this. Any help would be ...
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### Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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### 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 $A$, ...
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### Non empty interior in the image implies open map

I am looking at the proof showing that $L^2(0,1)$ is meager in $L^1(0,1)$. Define $B_n = \{f\in L^2 : \|f\|_2 \leq n\}$. With the continuous identity map $T:L^2 \rightarrow L^1$, if one of $T(B_n)$ ...
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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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### Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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### Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, ...
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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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### Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
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### Non differentiable solutions to $\partial_x f + \partial_y f =0$

This nice paper by Gilles Godefroy (in French) tells us the story of Baire's lemma. In 1896, Monsieur Baire was lecturing on analysis and carelessly gave the following exercise: to find all solutions ...
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### Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
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### x-section of closure of E of first category implies x-section of E nowhere dense

Let $E$ be a subset of first category of product space $X \times Y$. Why is the following true: if $(\bar E)_x \subset Y$ is of first category then it follows that $E_x$ is nowhere dense. $E_x$ ...
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### Fourier series converges “almost everywhere”

I'm reading "Fourier series" by Rajendra Bhatia. At one point, the author says: "[..]one can show the existence of a continuous function whose Fourier series diverges except on a set of points of the ...
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### Prove that the Set $P$ of algebraic polynomial is a first category set in $C[a,b]$

Prove that the set $P$ of algebraic polynomial is a first category set in $C[a, b]$ I know the definition of first category is countable union of nowhere dense sets. and further more I know that the ...
$\newcommand{\R}{\mathbf R}$ Claim. Let $m\neq n$ be positive integers. Can there exist a bijective continuous map $f:\R^m\to \R^n$? I think the answer is no and following is my argument. Purported ...