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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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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2answers
81 views

Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
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1answer
102 views

Filter, which does not have the Baire property

I do not understand the following proof of the following theorem. It is part of Theorem 4.1.2 (Talagrand theorem) in the book Tomek Bartoszyński and Haim Judah: Set theory. On the structure of the ...
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1answer
104 views

Meager subset of $2^\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
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1answer
38 views

Baire space using extended metric?

Consider the set $C^1(\mathbb{R},\mathbb{R})$ of continuously differentiable functions on $\mathbb{R}$, endowed with the extended $C^1$ norm $\|f\|_{C^1} = \sup_{x\in \mathbb{R}} |f(x)| + \sup_{x\in \...
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1answer
53 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
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2answers
31 views

Paths in a connected subset of $\mathbb R^n$

Prove that every connected subset $X$ of $\mathbb R^n$ with more than one point has the continuum cardinality. In order to use Baire Lemma, I would like to demonstrate that there is a compact (or ...
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0answers
19 views

$f$ is of Baire class $\xi$ implies existence of a topology such that $f$ is continuous with respect to that topology

Suppose that $(X,\tau)$ are $Y$ are Polish spaces and a function $f:X\rightarrow Y$. Show that $f$ is of Baire class $\xi$ if and only if there is a Polish topology $\tau^{\prime} \supset \tau$ with $\...
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0answers
10 views

Show that $f \cdot \chi_F$ is of Baire class $\xi$

A real-valued function $f : X \rightarrow \mathbb{R}$ is of Baire class $\xi$ if the sets $\{ x \in X : f(x)<c \}$ and $\{ x \in X : f(x)>c \}$ are in $\sum_{\xi+1}^0$ for every $c \in \mathbb{R}...
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1answer
20 views

Why for every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets?

$f$ is a real-valued baire class $\xi$ function if $\{ f < c \}$ and $\{ f >c \}$ are in $\sum_{\xi +1}^0$ for every $c \in \mathbb{R}$. In the proof of Proposition $5.2$ page $24$, we have ...
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1answer
32 views

Shift of first category set in a compact metric space.

Question. If $X$ is a homogeneous compact metric space, and $F=\bigcup _{n\in\omega}F_n$ is a countable union of closed nowhere dense subsets of $X$, then is there a homeomorphism $\varphi:X\to X$ ...
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1answer
42 views

Proof that a space is a Baire space.

Let $(Z,T_2)$ be a subspace of $R^2$ given by $Z = \{\langle x,y\rangle : x,y \in \Bbb R, y > 0\} \cup \{\langle x,0\rangle : x \in \Bbb Q\}$. Show that this is a Baire space. It is suggesting to ...
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1answer
59 views

Borel set as union of $G_\delta$ and countable set

What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?
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3answers
87 views

What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
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0answers
57 views

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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2answers
31 views

Prove that there is no countable base to a complete normed space

Defenition(to make sure we are talking about the same thing) : A base $B=\{b_1,b_2,\dots \}$ to a normic space $X$ is a group of elements from $X$, that satisfies the following condition : $\...
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0answers
25 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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2answers
45 views

Why if $p (x_1, \dots, x_n)$ is polynomial on $\mathbb{R}^n$, then $p (x) \neq 0$ is satisfied by open dense set?

I have problems in seeing what exactly is the all point of first category and second category sets. Finally, I've found a reference (Bredon's "Topology and Geometry") that introduces the topic in a ...
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0answers
17 views

Proving Banach space is finite in Baire space [duplicate]

Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, finite....
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1answer
51 views

$X,Y$ metric spaces , $X$ complete , $Z$ is Hausdorff , $f,g:X \times Y \to Z$ continuous in each variable and coincide on a dense subset , is $f=g$?

Let $X,Y$ be metric spaces , $X$ be complete and $Z$ be a Hausdorff space ; let $f,g:X \times Y \to Z$ be functions such that each of $f$ and $g$ is continuous in $x \in X $ for each fixed $y \in Y$ ...
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1answer
51 views

Real Analysis, Folland Problem 5.3.32 The Baire Category Theorem

Problem 5.3.32 - Let $\lVert x \rVert_{1}$ and $\lVert x \rVert_{2}$ be norms on the vector space $\mathscr{X}$ such that $\lVert x \rVert_{1}\leq \lVert x \rVert_{2}$. If $\mathscr{X}$ is complete ...
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1answer
64 views

There exist meager subsets of $\mathbb{R}$ whose complements have Lebesgue measure zero

The Baire Category Theorem - Let $X$ be a complete metric space a.) If $\{U_n\}_1^\infty$ is a sequence of open dense subsets of $X$, then $\bigcap_1^\infty U_n$ is dense in $X$. b.) $X$ is not a ...
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0answers
34 views

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 ...
2
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1answer
30 views

Lebesgue null set with meagre complement

There exists a $\lambda$-null comeagre subset of $\mathbb{R}$. I tried to find an example in terms of Cantor sets. Let $C \subseteq [0,1]$ be the ternary Cantor set. Define $$ S = \bigcup_{x \in \...
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2answers
112 views

Prove that $\mathbb{R}^2$ cannot be a subset of the union of a countable collection of lines in $\mathbb{R}^2$

I'm not really sure about this. I think the proof relies on the theorem: If $U$ is a subset of $A$, and $U$ is uncountable, then $A$ is uncountable. The problem statement says a collection of lines ...
2
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1answer
37 views

Equivalence between two comeager sets related to free groups

Let $G$ be a non discrete Polish group. For every $n\ge 2$ equip $G^n$ with the product topology. Saying that $F_n=\{(g_1,\dots,g_n)\in G^n: \{g_1,...,g_n\}$ freely generates a free subgroup of rank ...
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2answers
58 views

Is a comeager set also comeager with respect to any closed set containing it?

Let $X$ be a topological space. Let $Y,Z$ be two subsets of $X$. If $Z$ is comeager in $\overline Y$ and $Z\subseteq Y$ then $Z$ is comeager in $Y$ too?
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2answers
50 views

Why are theorems such as the Baire Category Theorem proved for $C([0,1])$ and not more general spaces?

In analysis I see that the proof of the Baire Category Theorem is proved for the set of all continuous functions on $[0,1]$, $C([0,1])$. However, I was wondering if the BCT would also hold for the set ...
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0answers
31 views

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 ...
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2answers
38 views

completeness and the Baire category theorem

I am studying the baire category theorem and trying to find a counterexample. This theorem says that a non-empty complete metric space can not be the countable union of nowhere-dense closed subsets ...
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1answer
36 views

on a countably union of $F_\sigma$ sets

Let $F$ be a countably union of $F_\sigma$ sets, say $F_i$ for $i\in \mathbb N$. Is it true that if $F$ is not meager then there is an $F_i$ with non-empty interior? For me it's true but I can't find ...
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1answer
53 views

Every comeager set in a perfect Polish space contains an uncountable dense $G_\delta$ set

Let $X$ be a perfect Polish Space. Prove that every comeager contains an uncountable dense $G_\delta$ set. It's known that every perfect Polish Space has cardinality $2^{\aleph_0}$. It's easy to ...
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1answer
50 views

The intersection of comeager sets in a Baire Space [closed]

Is it true that in a Baire Space the intersection of two comeager sets is not empty? If yes, is the intersection comeager too?
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1answer
29 views

is the complement of first category is always second category

is the complement of first category is always second category in general space( which is not complete). I think it is true only if the space isx
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1answer
21 views

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G.

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G. I tried by contradiction but could not figure it out. I found that we can use following result ...
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2answers
49 views

Zero-dimensional separable metric spaces

I have to prove that every separable metric space, which is zero-dimensional is isomorphic to a closed subset of the Baire space. Maybe I can use the Baire category theorem, but I don't know how.
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0answers
88 views

Proof of Invariance of Domain Using Baire Category and Local Homology

$\newcommand{\R}{\mathbf R}$ 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 ...
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3answers
113 views

If G is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in R. is it true in general metric space?

If $G$ is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in $\mathbb R$. is it true in general metric space? I know as $G$ is open and singleton set $\{...
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1answer
21 views

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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1answer
35 views

Empty interior, equivalent definitions from Munkres.

The Munkres book states the following definition: Recall that if $A$ is a subset of a space $X$ the interior of $A$ is defined as the union of all open sets of $X$ that are contained in $A$. ...
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1answer
31 views

Continuous image of a F-space

Let $X$ be a $F$-space, i.e. there exists complete metric $d$ on $X$ such that $d(x,y)=d(x+z,y+z), \ x,y,z\in X$. Suppose that we have a normable vector space $Y$, which is of first Baire category. ...
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0answers
89 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in B}\bigcap_{t\in\mathbb{R}}F_{-h(t)...
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2answers
139 views

Baire Category Theorem & The Real Numbers

I am taking a Real Analysis unit at University and the topic of the Baire Category Theorem is prevalent in all of the course, however I'm actually, embarrassing stuck right at the start of the ...
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2answers
356 views

Contradiction to Baire's Theorem?

My book states a version of Baire's Theorem as If $\{G_1,G_2,G_3,\ldots\}$ is a countable collection of dense, open sets, then the intersection $\bigcap_{n=1}^\infty G_n$ is not empty. I am ...
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1answer
74 views

The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
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1answer
44 views

Is the product of second category spaces second category?

The following is an exercise of Gemignani's Elementary topology. (I'm studying for an exam.) Suppose $Y$ and $Z$ are subspaces of some space $X,\tau$ and $Y$ and $Z$ are both of the second category in ...
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0answers
66 views

Prove that there is a dense subset of X on which $f$ is continuous.

Let ${\left(f_n\right)}$ be sequence of continuous function on a complete metric space $X$ which converges point-wise to a function $f$ then prove that there is a dense subset of $X$ on which $f$ is ...
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1answer
39 views

Problem seeing how Baire's theorem applies in a proof of open mapping theorem

I cant see which version and how they use Baires theorem to get that atleast on $MB_{n}$ is dense in some open set. Any version of Baires theorem needs open or closed sets. I can get neither on the $...
2
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1answer
46 views

$G_\delta$ 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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1answer
89 views

Nowhere dense subset of $L^1$

Why is $B_n = \{f \in L^1 : \int |f|^2 < n \}$, $n \in \mathbb{N}$ a nowhere dense subset of $L^1$? Please provide a proof without assuming that $L^2 \subsetneq L^1$. Clarification: $L^p$ here ...