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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Why empty set? (consequences of Baire's theorem)

I did not understand the proof of Theorem 5.13 of Rudin, [Real and Complex Analysis]. See next. In a complete metric space X which has no isolated points, no countable dense set is a $G_{\delta}$. ...
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Proving a linear function is bounded using the Baire category theorem (or its consequences)

This is a problem from Folland. Let $\mathcal{X}, \mathcal{Y}$ be Banach spaces. If $T : \mathcal{X} \rightarrow \mathcal{Y}$ is linear and $f \circ T \in \mathcal{X}^*$ for all $f \in ...
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44 views

Second category subset vs subspace

Let $(X,T)$ be a topological space. When we say $A$ is first category in $(X,T)$ we mean that it is the union of (or is covered by) countably many sets which are nowhere dense in $(X,T)$. $A$ is ...
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Non differentiable solutions to $\partial_x f + \partial_y f =0$

This nice paper by Gilles Godefroy (in French) tells us the story of Baire's lemma. In 1896, Monsieur Baire was lecturing on analysis and carelessly gave the following exercise: to find all solutions ...
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62 views

Implication of Baire Category Theorem

The statement that the countable intersection of open dense sets is dense is equivalent to the statement that the countable union of nowhere dense sets contains no balls. When proving the ...
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Application of the Baire category theory

Definition: A set $M\subset X$ is called "of first category" if it is countable union of nowhere dense sets. Otherwise its called "of second category". I want to see whether the following sets are ...
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123 views

Compact Hausdorff space is of second category

Let $X$ be a compact Hausdorff space, prove $X$ is of second category. I found a proof of this theorem in the case of locally compact Hausdorff spaces. Let $E_n$ be open and dense in $X$, locally ...
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107 views

Differentiable function with a set of critical points of second category.

I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words: Let $U \subset \mathbb{R}$ open. Is there a differentiable ...
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102 views

If $X$ is complete, then there is no continuous and open $\,f:X \to \mathbb{Q}$

I've encountered the following question and got stuck : There is no continuous and open mapping $\,f:X \to \mathbb{Q},$ where $X$ is a complete metric space. I thought it had something to do with ...
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1answer
37 views

Question regarding the proof that every hamel basis of an infinite space is uncountable

I am reading the following question: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. And I am wondering why $$X=\bigcup_{n\in \mathbb N}X_n$$ ...
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213 views

Showing the sum of two sets contains an interval - Baire's Theorem?

If $E$ and $F$ are measurable subsets of $\mathbb{R}$ and $m(E), m(F) >0,$ then $E+F$ contains an interval. The path to the standard solution to this is built on the notion that a measurable ...
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43 views

Infinite intersection between a arbitrary set of integers and a set of floor powers

Let $E$ be an infinite set of positive integers, proves that there is a $\alpha \in \mathbb{R}$ such that $\{\left \lfloor \alpha^k \right \rfloor ;k \in \mathbb{N} \}\cap E$ is infinite. I have two ...
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1answer
65 views

Equivalence of Forms of Baire Category Theorem

I am trying to show the equivalence of two forms of the Baire Category Theorem. These are the two statements: Let $(X,d)$ be a complete metric space. Let $U_n$ be a dense, open set for each $n \in ...
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3answers
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baire category and the union of dyadic balls of rational center

Suppose that $\{r_n\}_{n=0}^\infty$ is an enumeration of $\mathbb{Q}^N$ and $U = \bigcup_{n=0}^\infty B(r_n,2^{-n})$. We can use a trivial measure theory argument to prove that $U \neq \mathbb{R}^N$. ...
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For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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49 views

Convergence of a sequence holomorphic functions

Let $f_n\in\mathcal{O}(\Omega)$ be a sequence of holomorphic functions, s. th. $f_n\rightarrow f$ pointwise in $\Omega$. Show that exist open and dense set $\Omega'\subset \Omega$ such that $f_n$ is ...
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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 ...
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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
30 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
34 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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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
371 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. ...
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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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38 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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241 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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35 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) ...
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1answer
50 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 ...
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1answer
64 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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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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453 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 ...
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139 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, ...
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1answer
603 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. ...
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1answer
80 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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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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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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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
225 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 = ...
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1answer
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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
129 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
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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 ...
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1answer
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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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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
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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
84 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?
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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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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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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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119 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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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 ...
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59 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!