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.

learn more… | top users | synonyms

0
votes
1answer
67 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 ...
2
votes
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 ...
0
votes
1answer
51 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\...
0
votes
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 \...
0
votes
2answers
30 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 ...
63
votes
6answers
8k views

Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
0
votes
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 $\...
0
votes
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}...
0
votes
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 ...
8
votes
1answer
234 views

Can a meager linear subspace be written as a countable increasing union of nowhere dense subspaces?

Let $X$ be a separable Banach space. In this question, "subspace" means a linear subspace, not necessarily closed. Suppose $E \subset X$ is a subspace which is meager, so that we can write $E = \...
2
votes
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 $\varphi:X\to X$ ...
3
votes
2answers
58 views

Analogy for Baire categories?

I'm looking for an analogy to grasp the intuitive notion of size that Baire categories on $\mathbb{R}$ provides. For instance, the cardinality of a subset of $\mathbb{R}$ provides a notion of size in ...
1
vote
1answer
40 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 ...
2
votes
1answer
20 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 $G_{\delta}$...
0
votes
1answer
57 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?
0
votes
1answer
94 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using the Baire category theorem? And which version of the theorem ...
2
votes
3answers
84 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, ...
4
votes
0answers
55 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, ...
0
votes
1answer
71 views

Baire Category theorem - Determining First or Second Category

I don't have a mathematician background (I am an engineer). I understand some concepts but the Baire Category theorem and the ideas of first and second category are still very abstract for me. I have ...
4
votes
2answers
243 views

Baire's property iff first category has dense complement.

Show that $(S, d)$ has Baire's property iff every set of first category has a dense complement. A set is of first category if it is a countable union of nowhere dense sets. First Category Baire's ...
0
votes
2answers
29 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 : $\...
1
vote
0answers
24 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 ...
0
votes
1answer
20 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 ...
0
votes
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 ...
0
votes
2answers
75 views

how to show boundary of either open or closed set is nowhere dense.

how to show boundary of either open or closed set is nowhere dense. i think we need to use baire category thm? countable intersection of dense, closed set is once again a dense, closed (and nonempty)...
21
votes
1answer
6k views

Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
1
vote
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....
1
vote
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 $y \in Y$ ...
12
votes
2answers
1k views

$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$ [duplicate]

Can anyone help me with this problem? Let $f:[0,\infty)\longrightarrow \mathbb R$ be a continuous function such that for each $x>0$, we have $\lim_{n\to \infty}f(nx)=0$. Then prove that $\lim_{...
0
votes
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?
-1
votes
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?
1
vote
1answer
50 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 ...
1
vote
1answer
60 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 ...
0
votes
0answers
33 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
votes
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 \...
5
votes
2answers
108 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
votes
1answer
33 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 ...
4
votes
2answers
49 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 ...
0
votes
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 ...
0
votes
2answers
33 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 ...
0
votes
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 ...
-2
votes
1answer
51 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 ...
0
votes
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
0
votes
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.
0
votes
0answers
84 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 ...
3
votes
3answers
112 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 $\{...
1
vote
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)...
2
votes
1answer
34 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 $A$. ...
1
vote
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. ...
3
votes
2answers
136 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 ...