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 ...
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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 ...
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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 ...
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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{...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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_\...
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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 $...
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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 (\...
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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{...
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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 ...
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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 ...
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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 $X$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}\...
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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 $X$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $(\...
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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 ...
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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. (...
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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?
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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 "...
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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 ...
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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 ...
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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$ ...
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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 ...
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$\{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 ...
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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\...
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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\}=\...