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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Cannot understand some parts of proof for R be a countable union of closed sets

Im reading chapter9 Category, Real Analysis, Carothers, 1ed, talking about discontinuous functions of metric space. Here is a proof for a theorem that R be a countable union of closed sets,: Baire's ...
3
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2answers
61 views

Strong version of Baire Category Theorem

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense. In such spaces is the intersection of fewer than $2^\omega$-many open ...
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1answer
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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?
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1answer
27 views

Two proofs with possibly Baire category theorem about completness.

I'm working with completness right now and I've come across two interesting problems. In my opinion they are worth a little bit attention . a) Let $K$ be closed subset with empty interior on ...
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1answer
29 views

Baire category theorem in use on a plane

Let $F\subset\mathbb{R}$ be a closed nowhere dense set. One must show there exists $(a,b)\in S^1$ for which $b\neq qa+c$, for all $q\in\mathbb{Q},c\in F$. It's my second question concerning Baire ...
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2answers
770 views

Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty

The following is problem 3.22 from Rudin's Princples of Mathematical Analysis: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove ...
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3answers
213 views

Proof that every closed subset of $\mathbb R$ is finite or countable or continuum.

I want to prove that every closed subset of $\mathbb R$ is finite or countable or continuum. I know that for arbitrary subset we can not make similar statements - because of continuum hypothesis. ...
3
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1answer
35 views

For $f :\mathbb R \to \mathbb R $, there exists an $(a,b)$, such that $f$ is bounded on a sequence with limit $x$, for all $x\in(a,b)$

I want to prove the following. Let $f : \mathbb R \mapsto \mathbb R $. Show that there exists an interval $(a,b) \in \mathbb R $ and $c >0 $: such that for any $x \in (a,b) $ there is a sequence ...
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20answers
10k 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 ...
0
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1answer
45 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 ...
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1answer
33 views

If $\mathbb{R}=\bigcup_{n=1}^{\infty}E_n$, then the closure of some $E_n$ contains an interval

I'm working on some problems in Carothers' Real Analysis. I just started the section on the Baire Category Theorem. Thus far Carothers has given the Baire Category Theorem for $\mathbb{R}$. Prove ...
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0answers
226 views

Using Baire Category Theorem to prove $\mathbb{R}^2\not\cong\mathbb{R}^3$.

How can we prove $\mathbb R ^ 2$ is not homeomorphic to $\mathbb R ^3$ using Baire Category Theorem? Here is a standard proof of this fact using algebraic topology. Note that $\mathbb{R}^{3}-\{x\}$ ...
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2answers
32 views

Baire related problem

Let $f:\Bbb R\to \Bbb R$, $f$ in $C^{\infty}$. Suppose that for all $x \in \Bbb R$, there exists an integer $n$ (which depends on $x$) such that $f^{n}(x) = 0$ ($f^{n}$ is composing $f$ $n$ times) ...
2
votes
1answer
38 views

Non-decidable $\Pi^0_1$ (effectively closed) classes

Are there non-decidable $\Pi^0_1$ (effectively closed) classes? According to a draft of Effectively closed sets by Cenzer and Remmel, the class $$ P = \{ 0^n1^\omega \mid n \in B\} $$ is a ...
1
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1answer
53 views

Is the union of all $l^p$ spaces meagre in $l^\infty$?

Is the union of all $l^p$ spaces meagre in $l^\infty$? i.e. is $ \bigcup_{p=1}^\infty l^p$ meagre? I am revisiting this variety of math after a long break so help is appreciated. Please correct ...
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2answers
54 views

Is $c_0$ meagre in $l^\infty$?

I have been reading up on the meagre sets sans an instructor after a 10 year break from this sort of mathematics and I would like to test my understanding. I want to know if the following is true (and ...
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2answers
422 views

Every space is “almost” Baire?

There is this theorem called the Banach category theorem which states that in every topological space any union of open sets of first category is of first category. Now doesn't this imply that every ...
3
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2answers
107 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
2
votes
1answer
288 views

In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?

I was playing with Baire's Theorem, and seemed to deduce the following: In a complete metric space $X$ that has no isolated points, any countable intersection of open dense sets is uncountable. ...
0
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1answer
52 views

A homeomorphism of a topological space with itself maps a set into one of the same category

Prove: If $h$ is a homeomorphism of $S$ onto $S$ and if $E\subset S$, then $E$ and $h(E)$ have the same category in $S$. Rudin, Functional Analysis, 2/e, p.43. (My own answer follows below.)
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1answer
34 views

Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...
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2answers
40 views

Is the complement of a set 1st category set $X$ of 2nd category?

If I have some metric space $M$ and find that $X \subset M$ is of 1st category (resp. 2nd category), is the complement of $X$, $X^c$ of second cateogry (resp. 1st category)? Thanks!
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1answer
31 views

How can the Baire Category Theorem be used to show a point in a complete metric space has some particular property?

My book gives a very tantalizing yet brief note that the Baire Category Theorem can be used to show that a point $x$ in a complete metric space $M$ has a particular property $P$. It states: "A ...
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1answer
200 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 = ...
0
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1answer
37 views

A set that is a countable intersection of open and dense sets but not open.

We know that Baire Category Theorem implies that in a complete metric space, the countable intersection of open and dense sets is nonempty and actually dense itself. But it is clear that a countable ...
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1answer
113 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 ...
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1answer
2k views

A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem

When I google "baire category theorem", I get a link to Ben Green's website. And at the end of the paper, he mentioned such a classic problem: Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a ...
8
votes
2answers
807 views

$\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$

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 ...
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1answer
98 views

Why is the subspace of polynomials with degree $\leq$ n nowhere dense in $\mathbb{R}[X]$?

There's a popular application of Baire's Category Theorem that shows that $\mathbb{R}[X]$ (the space of all polynomials with real coefficients) is not complete by showing it is a countable union of ...
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0answers
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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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1answer
46 views

Is complement of a 1st category set in a 2nd category space, dense?

Let $X$ be a second category space and $Y$ be a first category subspace of $X$. Is $X \setminus Y$ dense in $X$?
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1answer
54 views

Baire's Theorem proof regarding points revisited

My first question on this point was not answered. Here is the first part of Shilov's proof of Baire's theorem (not an exact lift from the book as I avoided mathematical symbols). I am trying to be ...
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1answer
59 views

B meager, not empty, not open implies complement is dense

Let X be topological space. The subset B is meager, not open and non-empty. How can you prove that the complement is dense?
0
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1answer
51 views

Baire category theorem to show a set is dense.

Consider $A_j$ a sequence of subsets of $[0,1]$ s.t. for each $N\geq 1$, $\bigcup_{j=N}^\infty A_j$ is open and dense in $[0,1]$. If $S$ is the set of points $x \in [0,1]$ s.t. $x \in A_j$ for ...
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0answers
51 views

how do i prove that the the set of irrationals cannot be a countable union of closed subsets? [duplicate]

Let $\mathbb{R}$ be equipped with the standard topology. Let $E$ be the set of irrational numbers. How do i prove that $E$ is not a countable union of closed subsets, using Baire Category Theorem?
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0answers
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Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
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1answer
123 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 ...
3
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1answer
108 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 ...
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0answers
44 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
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1answer
55 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
148 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 ...
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0answers
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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
76 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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1answer
121 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
211 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 ...
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1answer
44 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 ...
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2answers
56 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 ...
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3answers
150 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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3answers
62 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 ...
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1answer
38 views

Baire: Equivalent Statements II

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