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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1answer
67 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 ...
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0answers
17 views

Uniform boundedness… [duplicate]

Let $\{f_n\}$ be sequences of real valued continuous functions on $\Bbb R$ that is convergent pointwise on $\Bbb R$. Show that there exists $M>0$ and interval $I\subset \Bbb R$ such that ...
3
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1answer
73 views

A Baire category question

Let ${f_n}$ be a sequence of real valued continuous functions converging pointwise on $\Bbb R$. Show that there exists a number $M>0$ and an interval $I \subset \Bbb R$ such that $\sup\{ |f_n(x)|:x ...
5
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0answers
194 views

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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0answers
22 views

If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
3
votes
1answer
41 views

Linear functional in Banach space

Let $X$ be a Banach space, $(f_n)\in X^{*}$ a sequence with $f_n\neq 0 $ $ \forall n\in \Bbb N$. Show that there is a $x\in X$ such that $f_n(x)\neq 0 $ $\forall n$. Need some help. Thank you!
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1answer
41 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
2
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0answers
23 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 ...
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1answer
23 views

Equivalence of Baire Space definitions

I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent. Defn1: Any topological space X such that the intersection of any countable ...
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0answers
23 views

Kuratowski-Ulam Theorem

I am looking for a proof of the Kuratowski-Ulam Theorem: Let $X,Y$ be Baire spaces, $Y$ be second countable and let $R \subseteq X \times Y$ have the Baire property. Then $\{ x \in X \ | \ R_x ...
0
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0answers
65 views

A wonderful property of real line!

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
0
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0answers
22 views

Proof similar to that of Baire's theorem

Sometimes, I read proofs that are very similar to the proof of Baire's theorem, but which I can't directly simplify by using Baire's theorem. I'll give an example: Let $M$ be a complete metric or ...
0
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1answer
31 views

Application of Baire's theorem

Let $f: (a,b) \rightarrow \Bbb R$ be a differentiable function in $(a,b)$. Calculate the pointwise limit of: $$f_n(x)=n(f(x+1/n)-f(x)), x\in(a, b-1/n). $$ Let $E_n$ be a countable ...
0
votes
0answers
15 views

No Baire measurable reduction from $\mathbb{E_0}$ to the diagonal of a second countable Hausdorff Space

On $2^{\mathbb N}$ we define the equivalence relation $x \mathbb{E_0} y \text{ iff } \exists m \in \mathbb N \forall n \ge m: x(n) = y(n)$ and notice that every $\mathbb E_0$-invariant set with the ...
0
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2answers
36 views

Baire space but not locally compact

I need an example that is a Baire space but not locally compact. I think, $\mathbb{R}^ \mathbb{R}$ is such an example. $\mathbb{R}^ \mathbb{R}$ is not locally compact. But I could not proof that it ...
2
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3answers
39 views

Meagre Sets: Algebra

Let meagre subsets be defined as: $A\text{ meagre}\iff A=\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ with }\overline{A_k}°=\varnothing$ Then it satisfies: $B\subseteq A\text{ meagre}\Rightarrow ...
1
vote
1answer
30 views

Baire: Equivalent Statements II

Moreover, why does it follow for Baire Spaces and why is it strictly weaker that: $X=\bigcup_{k=1}^\infty A_k\quad\Rightarrow\quad\exists k_0\in\mathbb{N},x_0\in X: A_{k_0}\in\mathcal{N}_{x_0}$ ...
2
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2answers
71 views

Baire: Equivalent Statements

How to proof that these statements are equivalent: Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. ...
1
vote
1answer
40 views

Prove that a set is dense and of the first category in $L^2(T)$

Define the Fourier coefficients $\hat{f}(n)$ of a function $f\in L^2(T)$, $T$ is a unit circle, by: $\hat{f}(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(e^{i\theta})e^{-in\theta}d\theta$ Let ...
3
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2answers
134 views

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
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1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
1
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1answer
50 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
4
votes
1answer
42 views

Baire Category Theorem proof in Gamelin Greene - how do they shrink the closure of open ball

I am confused by a step in the Gamelin and Greene proof of the Baire Category Theorem. Here is the start of the proof. Theorem: Let $\{U_n\}_{n=1}^{\infty}$ be a sequence of dense open subsets of a ...
0
votes
1answer
60 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
2
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1answer
30 views

Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
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1answer
55 views

Baire category related question

Let $A_n$ be a sequence of closed sets of $\mathbb{R}$ such that $[a,b]\subseteq \cup A_n$, for some $a<b $. Show that at least one of the set $A_n$'s contains an interval. Thanks in advance,
0
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1answer
51 views

Oxtoby - Thm 6.2 - There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category

I am reading the book Measure and Category of Oxtoby. In Theorem 6.2 it is stated that There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category As in the picture below. My ...
6
votes
2answers
120 views

Who are the measurable sets in $\mathbb{R}$

Is there any Characterization for all measurable sets in $\mathbb{R}$? Can I say that a set is measurable if an only if it has the property of Baire? (differs from an open set by a first category ...
0
votes
1answer
79 views

Why any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset homeomorphic to $[0,1]$

as a continuation to my question here: Is cantor set homeomorphic to the unit interval? I can't see how can it is be true that any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset ...
0
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1answer
49 views

Baire's theorem proof

> A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets. I don't understand the following point the proof: we assume that $X$ is the countable union of ...
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3answers
79 views

Is a Baire Space necessarily complete?

A complete metric space a Baire space. But is a Baire space necessarily complete?
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1answer
45 views

Baire Category Theorem: What should we really prove there?

I am reading about the Baire Category Theorem in Jech's book on set theory. 4.8: Baire Category Theorem: Let $D_0,D_1,\dots,D_n,\dots$, $n \in \mathbb{N}$, be open dense subset of $\mathbb{R}$. Then ...
0
votes
1answer
87 views

No infinite-dimensional $F$-space has a countable Hamel basis

If $X$ is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove that $X$ is of the first category in itself. Prove that therefore ...
1
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1answer
40 views

definition of Baire first category set

Suppose that $A$ is of the first category. Is it possible to write $A$ as a countable union of sets that are not necessarily nowhere dense? I mean after all a first category set is defined such that ...
3
votes
1answer
95 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
2
votes
1answer
110 views

Examples of rare, meager and nonmeager sets in $\mathbb{R}$

Kreyszig Functional Analysis book presents the following definition. I'm trying to get some examples. (a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so ...
2
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0answers
44 views

Baire Category Theorem with $G_{\delta}$

If we have a subset $Y$ of a complete metric space $X$ such that $Y$ is a $G_{\delta}$, then how would I go to show that $Y$ satisfies the Baire category theorem? I am trying to get there by showing ...
7
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0answers
118 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 = ...
1
vote
1answer
109 views

Can somebody help me to understand this? (Baire Category Theorem)

Theorem $\mathbf{6.11}$ (Baire Category Theorem) Every residual subset of $\Bbb R$ is dense in $\Bbb R$. $\mathbf{6.4.5}$ Suppose that $\bigcup_{n=1}^\infty A_n$ contains some interval ...
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1answer
185 views

Proving the existence of a non-monotone continuous function defined on $[0,1]$

Let $(I_n)_{n \in \mathbb N}$ be the sequence of intervals of $[0,1]$ with rational endpoints, and for every $n \in \mathbb N~$ let $E_n=\{f \in C[0,1] : f \:\text{is monotone in}\: I_n\}$. Prove that ...
0
votes
1answer
60 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 Baire category theorem? and which of the theorem version is ...
2
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1answer
64 views

Genericity and category

This paper by Ambos-Spies and Mayordomo on the theory of algorithmic randomness introduces the notion of genericity saying that it is based on Baire category while the usual notion of randomness is ...
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0answers
33 views

Sets with the property of baire in proof of Poincare Recurrence Theorem

I have a problem with the proof of Thm 17.1 in Oxtoby's Category and Measure. In statement of the Poincare Recurrence Theorem he does not say anything about property of Baire, but in the proof he ...
0
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1answer
85 views

Any example of open set of first category?

In the book by Oxtoby, the chapter 16 is devoted to them and Banach Category Theorem. However I do not understand this chapter and parts of the following one. I know that I can define any topology I ...
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votes
2answers
29 views

Confusion about this assertion

I have seen this result true in general ? Every zero set is $G_\delta$-sets. As I know in Normal spaces, every closed $G_\delta$ set is zero set. Thx in advance
4
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1answer
135 views

$L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b) [duplicate]

We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$. But in (b) they ask to ...
3
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1answer
91 views

Measure spaces are proper subsets

I want to prove that $L^2$ is of the first category in $L^1$, thus I have to prove that $L^2$ is the countable union of nowhere dense subsets. The hint I get is: Take $g_n(x)=n$ for $0\leq ...
1
vote
1answer
91 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
0
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1answer
31 views

In a Baire space $X$, if an open set meets a nonmeager set, is the intersection nonmeager?

In a Baire space $X$, if an open set $U$ meets a nonmeager set $N$, is the intersection nonmeager? If they do not meet then it's false; take the upper half line of the reals. It does not meet some of ...
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1answer
103 views

Is this Baire theorem a special case of Baire category theorem?

I'm reading a chinese text book "Real Analysis" (by Zhou Minqiang), one of it's conclusion is "Baire theorem" For any $E\subset R^n$ is a $F_\sigma$ set: $$E=\bigcup_{k=1}^{\infty}F_k,$$ where ...