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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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\}=\...
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Baire-1 functions properties on a complete metric space $(X, d)$

Let be $(X, d)$ a complete metric space, $f_n : X \rightarrow \mathbb{R}$ a sequence of continuous functions and a function $f: X \rightarrow \mathbb{R}$ such that: $$\lim_{n \rightarrow +\infty} f_n(...
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Baire space and increasing union of closed subspaces

Let $X$ be a Baire space. Suppose there is an increasing sequence $C_1\subset C_2\subset \cdots $ of closed subspaces of $X$, whose set-theoretical union is $X$. Since $X$ is Baire, we know that some ...
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The set is meager if it has a cover of clopen meager sets

Let $X$ be a topological space such that there exists a collection of meager clopen sets $(C_i)_{i \in I}$ such that $X = \bigcup_{i \in I} C_i$. I want to prove that $X$ is then meager itself. As ...
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Does a $G_\delta$ set of second category always contain an $F_\sigma$ set of second category?

We know that any $F_\sigma$ subset of $\mathbb{R}$ of second category always contains a $G_\delta$ set of second category, because an $F_\sigma$ set has the property of Baire and therefore can be ...
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Countable sum of Baire of class 2 functions on $\Bbb R$

For every $k\in\Bbb Z$, let $f_k$ be a Baire 2 class function on $\Bbb R.$ Assume $\sum_{k\in\Bbb Z} f_k$ is convergent. Define $f:=\sum_{k\in\Bbb Z} f_k$ so $f$ is a function. Moreover, $f$ is a ...
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Subset with only digits $0$ and $1$ in their decimal expansion

Consider $[0, 1]$ with its Euclidean metric and the set $A ⊆ [0,1]$ of real numbers $x$ that contain only digits $0$ and $1$ in their decimal expansion. It is clear to me why this set is not dense : ...
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More characterizations of Baire space

We know there are a bunch of equivalent definitions of Baire space. For example: Given a metric space $M$. 1.The complement of 1st category set in $M$ is dense in $M$. For more equivalent definitions, ...
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Can we classify all topological space $(X, \tau) $ where every second category sets are Residual sets?

$(X, \tau) $ be a topological space. $A\subset X$ is Residual if $X\setminus A$ is of first category. In a Baire space, a Residual set is of second category. $A\subset X$ Residual, then $X\setminus A$ ...
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If a Banach space has a countably infinite basis, then it has finite dimension

I'm solving Ex 3.8.2. in Brezis' book of Functional Analysis. Could you have a check on my attempt? Let $(E, | \cdot |)$ be a Banach space. Let $(e_n)$ be a countable basis of $E$, i.e., each $x\in E$...
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Show that a set of continuity points has an open dense subset

Let $f_n:\mathbb{R}\to\mathbb{R}$ be continuous maps, $\forall n\in\mathbb{N}$, and $f_n(x)\to f(x),\forall x\in\mathbb{R}$. Let $\varepsilon>0$. Show that there there is an $N\in\mathbb{N}$ and a ...
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Why was the concept of first/second categories in metric spaces introduced?

Let $(X, d) $ be a metric space. $X$ is of first category if it can be expressed as a countable union of nowhere dense subsets. Otherwise, $X$ is called a metric space of second category. A first ...
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Uncountable subset $X$ of $\mathbb{R^{n}}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$.

$\textbf{Question}$: Show that there exist an uncountable subset $X$ of $\mathbb{R}^{n}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$. $\textbf{My Attempt}$: ...
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Show if a family of maps if bounded pointwise, then it is bounded uniformly on a dense subset of an open ball

Let $M',M$ be metric spaces with $M$ a Baire space. If any family $(f_i)_i$ of maps $M\to M'$ is bounded pointwise on $M$, then it is also bounded uniformly on a dense subset of an open ball in $M$. ...
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Question about dense subset of an Banach space

Let $E$ be an Banach space and $A \subset E$ an dense subset. It is possible to find a function $f:E \to \mathbb{R}$ such that, for each $x \in A$, $\lim_{t \to x} |f(t)|=\infty$? I don't know how to ...
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Simple question about Open Mapping Theorem's proof in Functional Analysis

Regarding the OMT's proof, I have a really simple question, but I can't found an acceptable answer for it, every source simply ommits it (for this motive I think it's really simple). So, let's state ...
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Baire space and Topological property

Show that the property of being a Baire space is a topological property . Baire space : In mathematics, a topological space X is said to be a Baire space, if for any given countable collection {An}...
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Complement of meagre set contains a dense $G_\delta$ set

While reading a Functional Analysis book, the autor stated, in a somewhat crude manner, that, if $X$ is a Baire space and $H \subset X$ is meagre, then $X \setminus H$ contains a dense $G_\delta$ set. ...
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What is the justification for these two steps in proof of Baire's theorem?

Below is the statement and proof of Baire theorem from the textbook I'm using. I have some difficulties understanding it and I would appreciate if someone can explain it in more details. In particular ...
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Is an NVS complete iff it is non meagre?

Let $V$ be an NVS. We know by Baires theorem that complete $\implies$ nonmeagre. What about the converse? Can we have a non complete space that is nonmeagre or not? For metric spaces the answer is yes ...
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Why are these two caracterisations of a meagre set equivalent?

In different courses, I have the following two caracterisations of a meagre set: (1) A is meagre iff $A=\cup_{d=1}^{\infty} N_d$ where $N_d$ are nowhere-dense sets. (2) A is meagre iff $A^C \subseteq \...
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2 answers
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An application of Baire Category Theorem

I am trying to prove a proposition that $BV[a.b]\cap C[a.b]$ equipped with the $||\cdot||_\infty$ is Baire 1 category set, which will tell us that $E=\{f:V(f)=\infty, f\in C[a,b]\}$ is a dense Baire 2 ...
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An interesting problem involving recursion

Given a continuous function $f:[0, 1] \rightarrow [0, 1]$. Here we denote $f^n(x) = f(f^{n-1}(x))$. For every $x_0 \in [0, 1]$ there exists $n \in \mathbb{N}$ such that $f^n(x_0) = 0$. Prove that $f^N(...
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1 answer
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How to construct a real valued function which is continuous only on a given $G_\delta$ subset of $\mathbb{R}$?

I know that the set of continuity of a function $f:\mathbb{R} \to \mathbb{R}$ forms a $G_{\delta}$ subset of $\mathbb{R}$. My question is, given any $G_{\delta}$ subset of $\mathbb{R}$ , how can I ...
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$A$ has property of Baire if and only if $A=B \sqcup Q$ where $B$ is a $G_{\delta}$ set and $Q$ is of first category

Over $\Bbb{R}$, A set $A$ with a property of Baire is defined in our notes as the symmetric difference $A=G\triangle Q$ where $G$ is open and $Q$ is of first category. I am asked to show that a set $A$...
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The image of bounded linear operator is the whole space or first Baire category [duplicate]

Let $V$ be a complete normed space and $W$ is normed space. Let $A \subset V$ be closed subspace. If $ T:V \rightarrow W$ is bounded linear operator, then we need to prove that $T(A)$ is either first ...
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Topological transitivity definition

Let $X$ be a Baire space, $f$ a map $X \to X$, $x$ a point of $X$, $\omega (x,f)$ its limit set, $Trans(f)$ the set of transitive points. In many sources (e.g. here) the following definitions of "...
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1 answer
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Proving a subset of $\mathbb R$ is $G_{\delta}$.

Using the Baire Category Theorem, I am trying to solve the following problem: Let $d(x) = \text{dist}(x, \mathbb Z)$ denote the distance from $x \in \mathbb R$ to the nearest integer. For $q \in \...
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Isolated point in a subset

Let $(X,d)$ a metric space. Let $A$ be a subset of $X$. A point $a\in A$ is isolated if only if there is $r>0$ such that $B(a,r)\cap A=\{a\}$. Let $a\in A$ be. I am trying to prove that $$\...
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Cardinality of set of Baire functions

I'm reading this paper of Sierpinski. At p.260 he says that it is well known that the set of all injective Baire functions (on the reals) is of cardinality $2^{\aleph_0}$, but he gives no reference. ...
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Proper subspace of $\mathbb{R}^{n}$ is nowhere dense.

How can we prove that if $V$ is proper subspace of $\mathbb{R}^{n}$, then $V$ is nowhere dense. My try: I will prove that any open ball $B(x,r)$ will not be contained in $V$. By contradiction, suppose ...
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Example of a metric space that is non-meager itself and not Baire?

Is there an example of a metric space that is not Baire and non-meager in itself? I know that that complete metric spaces Baire spaces and that Baire topological spaces are non-meager in themselves. ...
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Question 3.8 from Brezis' book of Functional Analysis

QUESTION: Let $E$ be an infinite-dimensional Banach space. Our purpose is to show that $E'$ equipped with the weak $^\star$ topology is not metrizable. Suppose, by contradiction, that there is a ...
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1 vote
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Using DC to prove Baire's category theorem in Munkres

From Munkres' Topology book: Theorem 48.2 (Baire category theorem). If $X$ is a compact Hausdorff space or a complete metric space, then $X$ is a Baire space. Proof. Given a countable collection $\{...
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3 votes
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On a sequence of continuous functions

I am working on the problem: Let $\{ f_n \}$ be a sequence of real-valued functions on $[0,1]$ defined by $f_0 = f \in C[0,1]$ and $f_n$ is an anti-derivative of $f_{n-1}.$ Suppose that for each $x \...
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Every continuous function on $[0,1]$ can be approximated uniformly by continuous nowhere differentiable functions [duplicate]

One can use the method in here or I guess I can approximate it by "zig-zag" function(a "zig-zag" function is a continuous piecewise-linear function,each of whose line segments have ...
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Set of differentiable functions.

The real analysis book that I had read mentions that "most continuous functions do not have derivatives at any point." The precise mathematical definition of "most" can be inferred ...
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1 answer
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Does there exist a dense subset $\ X\ $ of $\ \mathbb{R}\ $ and a real number $\ a\neq 0\ $ such that $\{\ x+a:\ x\in X\ \} =\mathbb{R}\setminus X\ ?$

Does there exist a dense subset $\ X\ $ of $\ \mathbb{R},\ $ and a real number $\ a\neq 0\ $ such that $\ \{\ x+a:\ x\in X\ \} = \mathbb{R}\setminus X\ ?$ Clearly our set $\ X\ $ must be totally ...
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1 vote
1 answer
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Showing that a set is residual in the metric space $(A,d_A)$

Let $D = [c,d]\times [c,d]\subseteq \mathbb{R}^2$ and let $A$ be the set of all closed subsets of $D$. For $a \in D$ and $B\in A,$ define $d(a,B) := \min\{d(a,b) | b\in B\},$ where the $d$ inside the ...
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