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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Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
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1answer
63 views

Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function such that for every $x \in \mathbb{R}$ there exists $n$ such that $f^{(n)}(x) = 0$, then f is a polynomial. I'm kind of lost on this ...
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1answer
80 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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2answers
44 views

Undestanding an assumption in Baire's theorem proof.

I was reviewing the proof of Baire's theorem I saw in class a few days ago, and there's an assumption that I didn't managed to see where it was used. I'll put here the proof given. Theorem (Baire) ...
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1answer
23 views

Is the closure of a meager set meager?

How to show that the closure of a meager set is meager? I tried like this: Suppose that it is not meager then $cl(A)$, where $A$ is a meager set in a metric space $(X,d)$ contains an interior point ...
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0answers
27 views

Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
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1answer
45 views

Category theorem

I don't have a mathematician background (I am engineer) I understand some concepts but still very abstract for me and I have to show the following: 1.- Of what category is the set of all rational ...
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0answers
61 views

A function continuous on rational points and discontinuous on irrational points [duplicate]

How to find function $f : \Bbb R \to \Bbb R$ such that $f$ is continuous on the rational numbers and discontinuous at irrational numbers? I was told to use the Baire Theorem to show that the set of ...
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1answer
71 views

Prob. 5, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Every compact Hausdorff space is a Baire space

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
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2answers
119 views

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let ...
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47 views

Q: Nowhere dense sets.

Given $X$ a metric space, $A\subset X$ a nowhere dense set. Show that every open ball $B$ contains another open ball $B_1 \subset B$ such that $B_1 \cap A = \emptyset$. EDIT: I modify my proof ...
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2answers
52 views

Show that a countable dense subset $D \subset X$ is not a $G_{\delta}$

Given $X$ a complete metric space with no isolated points and $D \subset X$ a countable dense subspace, show that $D$ is not a $G_{\delta}$. I am quite lost in trying to use the hypothesis of the ...
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1answer
53 views

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces Ok so I know I have to use Baire's Cathegory Theorem here. And I've done the following, lets suppose on the ...
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1answer
46 views

Requirements for the principle of uniform boundedness

The version of the principle of uniform boundedness as we stated it in the lecture seems wrong to me in multiple points. Here is how I would state and proof the principle in the terms we used in the ...
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1answer
42 views

intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
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0answers
30 views

Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
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1answer
21 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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0answers
44 views

How it can be possible? [duplicate]

I can't understand how it can be possible? Prove that the interval $[0,1]$ has a uncountable partition $\mathcal P$ such that each $D\in \mathcal P$ is uncountable and dense in $[0,1]$.
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1answer
56 views

Baire category theorem in a Banach space

For any two distinct $u,v$ in a countable dense subset of separable real Banach space $X$, let $S(u,v) = \{f \in Y \mid f(u)=f(v)\}$, where $Y$ is the dual space of $X$. Each of $S(u,v)$ is a proper ...
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1answer
29 views

Why this set is of the second category?

I'm watching Baire space on en.wikipedia.org, and find this conclusion. Here is an example of a set of second category in $\mathbb R$ with Lebesgue measure zero. $$\bigcap_{m=1}^\infty ...
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1answer
75 views

On the proof of Baire category theorem

I would like to ask about the proof of Baire Category theorem found on wolfram. The excerpt is as below: Baire's category theorem, also known as Baire's theorem and the category theorem, is a result ...
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0answers
53 views

Producing $\mathbb{R}$ with countable amount of sets?

Prove, that you can't "produce" $\mathbb{R}$ with countable amount of sets, which are nowhere dense(I am not sure I said this definition correct, with nowhere dense, I mean that $Int(\overline X) = ...
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1answer
84 views

If $f:\mathbb R \to \mathbb R$ is an additive function whose graph is $G_{\delta}$ in $\mathbb R^2$ , then the graph is closed in $\mathbb R^2$?

If $f:\mathbb R \to \mathbb R$ is an additive function i.e. $f(x+y)=f(x)+f(y) ,\forall x,y \in \mathbb R $ such that $G(f):\{(x,f(x)) : x\in \mathbb R\}$ is a countable intersection of open sets , ...
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1answer
23 views

Is it true that every 1st category subset of a 2nd category space has empty interior?

Let $X$ be a metric space. Are these conditions equivalent: Each set of the 1. category in $X$ has empty interior; $X$ is of the 2. category. It is obvious that $1 \Rightarrow 2$. Is it true that ...
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1answer
30 views

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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1answer
35 views

Question about the Baire space, $\sigma$-algebra and $\sigma$-ideal.

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$. Assume $X$ is second countable Baire ...
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1answer
60 views

Question about of Baire property and Baire space

In reading Kechris book. Please, I would like help with this proposition. For convencion we put for $A \subseteq X$, $$\sim A=X\setminus A$$ If $A$ is comeager in $U$, we say that $U$ forces $A$, ...
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0answers
25 views

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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1answer
28 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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0answers
56 views

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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1answer
39 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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1answer
50 views

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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2answers
93 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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1answer
100 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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1answer
27 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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1answer
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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

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
50 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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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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1answer
38 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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1answer
140 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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1answer
65 views

Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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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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2answers
149 views

Baire's property iff first category has dense complement.

Show that $(S, d)$ has Baire's property iff every set of first category has a dense complement. A set is of first category if it is a countable union of nowhere dense sets. First Category Baire's ...
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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
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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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3answers
140 views

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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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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3answers
218 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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1answer
49 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 ...