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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2
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1answer
19 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 ...
0
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1answer
54 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
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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
78 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
54 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
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1answer
69 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
242 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
27 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
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0answers
20 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
19 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
44 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
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2answers
72 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 ...
19
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, ...
1
vote
1answer
50 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 ...
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
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1answer
49 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
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1answer
47 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
58 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
31 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
29 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
93 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
32 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
30 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
32 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
28 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
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2answers
46 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
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0answers
68 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
107 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
88 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 ...
2
votes
1answer
31 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
133 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 ...
2
votes
2answers
353 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 ...
0
votes
1answer
66 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 ...
111
votes
20answers
13k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
2
votes
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 ...
2
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0answers
62 views

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

Let ${(f_n)}$ 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 continuous. I ...
-1
votes
1answer
38 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 ...
1
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1answer
20 views

Second Baire class and Borel measurable

I am new to Baire class theory, but need it for one part of a project I am working on. I have seen it referenced that functions of second Baire class are Borel measurable. For example here in this ...
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vote
2answers
2k views

Is the union of two nowhere dense sets nowhere dense?

Is the union of two nowhere dense sets nowhere dense? Using the following definition: A nowhere dense set is a subset $E\subset X$ of a metric space (or topological space) $X$ such that ...
2
votes
1answer
42 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 ...
4
votes
2answers
181 views

Nowhere dense set…

Let $A_n$ be a subset of continuous functions on $[0,1]$ given by: $A_n$ = {$f∈C[0,1]$:there exists $x∈[0,1]$ such that $|f(x)−f(y)|≤n|x−y|$ for all $y∈[0,1]$}. Show $A_n$ is nowhere dense, and use ...
-1
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1answer
85 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 ...
7
votes
1answer
110 views

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 ...
0
votes
1answer
25 views

Closed set of 1st category

If $F$ is a closed subset of a complete metric space, is it possible for $F$ to be of the first category? This seems to lead to a contradiction. Since $F$ is a closed subset of a complete metric ...