Questions tagged [baire-category]

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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Prove that $[0,1]$ is not a disjoint union of a countable family of closed sets [closed]

This proof is probably too long, please don't downvote this. I'm just leaving it here in case somebody wants to actually verify its validity for me. I didn't use a hint in the book and decided to ...
Stanislav Veklenko's user avatar
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Nowhere Dense Set and Corollary to Baire's Theorem

Not Homework just Personal Study Reference text: Kolmogorov and Fomin - Introductory Real Analysis I am working on understanding why a complete metric space with no isolated points is uncountable. The ...
Mud's user avatar
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Is there a metric $d$ such that $(\mathbb{F}[x], d)$ is complete?

Denote $\mathbb{F}[x]$ (where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$) is the space consisted of polynomials with coefficients in the field $\mathbb{F}$. The question is whether there ...
Bakkune's user avatar
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f(tn) converges -> f(t) converges [duplicate]

Claim: Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a continous function, for which $\text{lim}_{n \rightarrow \infty}f(nt) = 0$ ($n \in \mathbb{N}$) holds for all $t \in [0, \infty)$. Then $\text{...
Jahi02's user avatar
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1 answer
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If $(X, \tau)$ is a Baire space, then $X \times [0,1]_{std}$ is a Baire space

I’m trying to prove the following claim: “If $(X, \tau)$ is a Baire space, then $X \times [0,1]_{std}$ is a Baire space” The only way I know how to prove this is by using a Theorem from John C Oxtoby, ...
obitobi_tobias's user avatar
7 votes
3 answers
642 views

Why topological spaces in Baire category theorem are required to be Hausdorff?

In Baire category theorem it says a locally compact Hausdorff space $S$ is second category. In the proof, it choose $V_1,V_2\cdots$ are dense open subset of $S$. $B_0$ is an arbitrary nonempty open ...
xyz's user avatar
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Baire Category in Exercise 4 of Chapter 2, Functional Analysis Rudin

In the next problem of Rudin's book, Functional Analysis, Let $L^1$ and $L^3$ be the usual Lebesgue spaces on the unit interval. Prove that $L^2$ is of the first category in $L^1$, in the following ...
Adrian's user avatar
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4 votes
1 answer
130 views

A corollary of Baire Category Theorem

I have learnt that the theorem says: Let $\{U_n\}_{n=1}^\infty$ be a sequence of dense open subsets of a complete metric space X. Then $\displaystyle\cap_{n=1}^\infty U_n$ is also dense in X. I also ...
Lee Wei Xuan's user avatar
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55 views

Let E be a non-meager subset of a Banach space, then E - E contains a neighborhood of zero.

I want to prove the statement: Let E be a non-meager subset of a Banach space, then E - E contains a neighborhood of zero. Here, E - E means the Minkowski difference of E with itself. The statement is ...
User84308209's user avatar
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Can a proper subspace of a Banach space be a countable intersection of open sets?

Let X be a Banach space and suppose E is a dense linear subspace which is a Gδ -set. Show that E = X. By the Baire Category theorem we can show that E is non-meager (second category), but I don't know ...
User84308209's user avatar
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51 views

When Should I Try to Use the Baire Category Theorem?

I've studied the statement and proof of the Baire Category Theorem for complete metric spaces and locally compact Hausdorff spaces, and I've used the theorem in an exercise to prove that that in ...
Nick A.'s user avatar
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A exercise assigned in the topology course

Let $\{x_n\}_{n=1}^{\infty}$ be an enumeration of $\Bbb Q$, $\{y_\lambda\}_{\lambda \in \Lambda}$ an enumeration of $\Bbb R\setminus\Bbb Q$. Proof: for any families $\{r_n\}_{n=1}^\infty,\,\{ \rho_\...
anyon's user avatar
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1 answer
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Alternative proofs of $L^q[0,1]$ is a (Baire) first category subset in $L^p[0,1]$

Let $1\leq p<q<\infty$, deduce from each one of the following statements that $L^q[0,1]$ is a (Baire) first category subset in $L^p[0,1]$: Let $p<r<s<q$ and $\beta=s/(s-1)$, we define $...
Sergio Ferrer's user avatar
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1 answer
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Brezis' Exercise 4.8

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 4.8 Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $X$ be a closed vector subspace of $L^1 (\...
Akira's user avatar
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Nowhere dense subsets of a dense subspace are nowhere dense in the whole space and vice versa

It seems, with the following lemma, the proposition at the bottom easily follows. If $Y\subset X$ dense. Then, for every nonempty $A\subset Y$, $\text{Int}_Y \left(\text{cl}_Y A\right)=Y\cap \text{...
user760's user avatar
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Equivalent definitions of Baire spaces

We say that a metric space $X$ is a Baire space if there is no open set $E$ such that $$E \subseteq \bigcup\limits_{n\geq 1} F_i,$$ in which each $F_i$ is a closed set with empty interior. Suppose ...
José Victor Gomes's user avatar
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Examples of dense $G_\delta$ sets in $L^p$ spaces

The Baire Category Theorem states that the countable intersections of open dense subsets of a complete metric space (called dense $G_\delta$ sets) are dense. Any open set is $G_\delta$, so any dense ...
Andrew Murdza's user avatar
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Baire category theorem and countable union of closed meagre sets in complete metric space

In my functional analysis course Baire category theorem is the following: Complete metric spaces are of second category. The following was stated as Baire Category theorem's consequence: Suppose $X$ ...
H-a-y-K's user avatar
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Why is this proof of the Baire Category Theorem (BCT) is incorrect?

The Baire Category Theorem states: Theorem: Let $X$ be a complete metric space. Suppose that each element of the sequence $\{U_n\}$ of open sets of $X$ is dense. It holds that $\cap^\infty U_n$ is ...
Kzards's user avatar
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0 answers
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The set of nowhere differentiable functions is generic (thus dense) in Hölder Space $\Lambda^\alpha(\mathbb{R})$ with $0<\alpha<1$

It is well known that the set of nowhere differentiable functions is generic (thus dense) in the space of continuous functions. (Partial) proofs can be found here and here. I would like to prove that ...
Petra Axolotl's user avatar
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1 answer
106 views

the strength of the axiom of choice used in forcing

I believe that Shelah's model gives a ZFC theorem of relative consistency results: $ZFC \vdash Con(ZFC + \text{there is a strongly inaccessible cardinal})\rightarrow Con(ZF+DC+ \text{all sets are ...
clhs's user avatar
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1 answer
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How can I show that $X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$ is not complete?

Let $X$ be a vector space of real polynomials, $$X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$$I want to show that $X$ is not compleate with respect to any norm on $X$. My idea was to ...
user1294729's user avatar
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1 answer
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Verify: The points of discontinuity of any $f: \mathbb R \to \mathbb R$ are a $F_\sigma$ set

Prove that the points of discontinuity of an arbitrary function $\mathbb R \to \mathbb R$ are a $F_\sigma$ set. Note: Proofs are available; this question is to verify and critique this proof and ...
SRobertJames's user avatar
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2 votes
1 answer
56 views

Verification of the converse of Baire Category Theorem

One way to state the Baire Category Theorem is as follows: If $X \neq \emptyset$ is a complete metric space, then $X$ is nonmeager as a subset of itself. Then it was mentioned that the converse, that ...
user57's user avatar
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Category of continuous functions with compact support in the space of continuous bounded functions and in itself

All functions here are from R to R and I use the supremum metric d. $C_c$= continuous functions with compact support and $C_b$ = continuous bounded functions. We say a metric space $(X,d)$ is of first ...
autodidacti's user avatar
1 vote
1 answer
55 views

Does there exist a nonmeager compact subset of an infinite-dimensional Banach space?

Let $ X $ be an infinite-dimensional Banach space and let $ K \subset X $ be compact. If $ K $ is contained in a finite dimensional subspace of $ X $, then $ K $ must be meager. However, there may ...
jzs's user avatar
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0 answers
23 views

If $B$ and $C$ are residual sets in $\mathbb{R}$, then $B-C = \mathbb{R}$

The definition of a "residual set" I'm using is "complement of a meager set", and a meager set is a countable union of nowhere dense sets. I've thought about approaching this in a ...
Dalop's user avatar
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0 answers
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Is the set $ A $ nowhere dense?

A part of an exercise I've been trying to solve asks to prove that the set $$ A = \left\{ \sum_{n=0}^{\infty} a_n x^n \, \Big| \, (a_n) \in \ell^1(\mathbb{N}) \right\} \subset C([0,1]) $$ is not ...
jzs's user avatar
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1 answer
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a question between the first category set and closed proper subspace

Assume $X$ is a complete norm space with $\mathbb{dim}~X=+\infty$, $E\subset X$ is a proper closed subspace. On one hand, I have known $E$ is a nowhere dense set, because any proper closed subspace of ...
fusheng's user avatar
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1 answer
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Countable Intersection of Sets of Second Category

We say a set is of first category if it is a union of countably many nowhere dense subsets. If a set is not of first category, we say it is of second category. Now I am thinking of a question. Suppose ...
Sam Wong's user avatar
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If $f(nc) \to 0$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$ [duplicate]

Suppose $f$ is a continuous function on $\Bbb R$. If $f(nc) \to 0$ as $n \to \infty$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$ I have solved the problem after extending the ...
Ri-Li's user avatar
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3 votes
0 answers
28 views

How to use non-standard analysis to prove Baire Category Theorem?

I'm caring about some questions of non-standard analysis. I have found the only book talking about Baire Category Theorem, which is the book of Siu-Ah Ng. But I think the proof in this book is not ...
Sigh酱's user avatar
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Show that $\mathbb{C}[z]$ is a first Category (as a subspace of All Square Integrable Holomorphic Functions on the Unit Disk)

Show that $\mathbb{C}[z]$ is a first Category (as a subspace of $A^2(\mathbb{D})$ = All Square Integrable Holomorphic Functions on the Unit Disk). I notice that $\mathbb{C}[z]=\cup_{n\geqslant 1}\...
Max Young's user avatar
1 vote
0 answers
19 views

If $X-M$ is a second category space then $M$ is a first category space

Let $X$ be a complete metric space, let $M \subseteq X$ such that $X-M$ is a second category space prove that $M$ is first category space. My attempt. We know that $X$ is complete metric space, so $X$ ...
erika21148's user avatar
0 votes
1 answer
68 views

Excercise 15 Rudin functional analysis chapter 2

I am self-studying the book function analysis of Rudin. I got stuck on the final passage of the following exercise. Suppose $X$ is an $F-$space (a topological vector space with a topology induced by a ...
Matteo Aldovardi's user avatar
1 vote
1 answer
157 views

Prove Baire category theorem for compact spaces.

Definition A subset $Y$ of a topological space $X$ is said nowhere dense if $\operatorname{int}(\operatorname{cl} Y)$ is empty. Now let be $X$ a compact topological space and thus let be $$ \mathcal ...
Antonio Maria Di Mauro's user avatar
1 vote
0 answers
35 views

Is the set of discontinuities of the following Baire first class function meager?

Let $E_1$ be a closed measure zero subset of the unit circle $C$. Moreover, assume that there exists a set $E_2\subset C$, such that $E_1\subset E_2$ and a bounded analytic function $f$ defined on the ...
TheGeometer's user avatar
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2 votes
1 answer
119 views

There exists a measurable subset of $\Bbb{R}$ without the property of Baire and a non measurable set with the property of Baire.

Theorem : There exists a measurable subset of $\Bbb{R}$ without the property of Baire and a non measurable set with the property of Baire. Lemma $ 1$: $ \Bbb{R}$ can be decomposed into disjoint union ...
Sourav Ghosh's user avatar
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Proof: If $(\displaystyle\bigcup_{i=1}^{\infty}A_i)^°\neq\emptyset$ there exists $i^*\in\mathbb{N}$ such that $(A_{i^*})^°\neq\emptyset$

I am preparing for my upcoming exam by doing some exercises and need help for the following task: Let $A_1,A_2,A_3,...$ be a sequence of closed subsets of $\mathbb{R^n}$ with the property that $(\...
Analysis_Mark's user avatar
3 votes
0 answers
144 views

The tent map system is transitive. Can we actually identify any of the infinitely many transitive points, however?

$\newcommand{\O}{\mathcal{O}}\newcommand{\G}{\mathcal{G}}\newcommand{\T}{\mathcal{T}}$TLDR; skip to the end of the preamble - we know that the tent map system is topologically transitive. However, do ...
FShrike's user avatar
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2 votes
1 answer
118 views

Example of function of Baire 3 and Baire 4

i'm looking for explicit examples of real-valued functions of the Baire third and fourth class, without using Borel-measurability but just using some characterization theorem of the previous classes. (...
Luca Cardarelli's user avatar
1 vote
0 answers
35 views

Product of two nonmeager sets

Let $A, B\subseteq \mathbb R$. Suppose both $A, B$ are non meager sets. Is is true that $A\times B\subseteq \mathbb R^2$ is non-meager? What if at leats one of them is co-meager?
Green Park's user avatar
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0 answers
85 views

Why Conway 13 base function is Baire two?

I'm writing my thesis about Baire Classes. I want to prove that the 13 base Conway function is in the Baire two class. I need a clear proof of this fact, a proof in which is described how to "...
Luca Cardarelli's user avatar
1 vote
1 answer
67 views

If we say a space is of the first category, do we by default mean in itself?

The definition of "being of first category) that I am using is provided below. I usually see this mentioned without specifying in which space it is meant. For example, Engelen says in one of his ...
Tereza Tizkova's user avatar
3 votes
0 answers
98 views

Application of Baire Category to show a set has non-empty interior

Suppose $X \subset \mathbb{R}$ is closed with the usual metric, and $\forall x \in \mathbb{R}, \, \exists n \,$ such that $\frac{x}{n} \in X$. I am trying to prove $int(X) \neq \emptyset$. I tried to ...
Smasher640's user avatar
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0 answers
63 views

Some consequences of the Baire Category Theorem

I read there are some consequences of the Baire Category Theorem in "The Integrals of Lebesgue, Denjoy, Perron, and Henstock" by Russel. A. Gordon. The statement is Corollary 5.3. Let $E$ ...
user136524's user avatar
0 votes
1 answer
122 views

Proof verification: countable intersection of dense $G_\delta$'s in a complete metric space is dense

I'm working through Kadets' book A Course on Functional Analysis and Measure Theory to brush up on my measure theory and wanted to spend a bit more time with the Baire category theorem. This is ...
csch2's user avatar
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2 votes
1 answer
108 views

$\{f \in L_2(\mathbb{T}): \sum_{n=-\infty}^{\infty}\hat{f}(n) \text{ is convergent}\}$ is a first-category Baire set in $L_2(\mathbb{T})$

Let be $L_2(\mathbb{T})=\{f:\mathbb{T}\rightarrow\mathbb{R} \ | \int_{\mathbb{T}}|f|^2d\mu <+\infty\}$, $\pi:\mathbb{R}\rightarrow \mathbb{T}$ the function that for every $x$ associates his ...
GoofyMushroom0's user avatar
0 votes
1 answer
131 views

Locally Baire Spaces

Let $X$ be a topological space such that $\forall x\in X$ $\exists U_x$ open neighborhood of $x$ such that $U_x$ is a Baire space. I have to prove that $X$ is a Baire space. Proof : Let $\{A_n\}_{n\in\...
Grace53's user avatar
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1 answer
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Can the set in which a sequence is unbonded be $\mathbb{Q}$?

Let be $f_n:\mathbb{R}\rightarrow[0, +\infty)$ as sequence of continuous and non-negative functions. Can $\{f_n\}_{n\geq1}$ exist such that: $$\{\ x \in \mathbb{R}\ |\ \sup_{n\geq1}f_n(x)=+\infty\}=\...
GoofyMushroom0's user avatar

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