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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Reference book on Kronecker set

I am self-reading the lecture notes Baire Category, Probabilistic Constructions and Convolution Squares by T.W. Korner. https://www.dpmms.cam.ac.uk/~twk/Baire.pdf In chapter 4, he mentioned the ...
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understanding a proof about the set of nowhere differentiable functions

Show that the set of nowhere differentiable functions is residual in $\mathcal{C}[0,1]$. It suffices to show its complement is first category. Below is part of a proof of this statement. In the proof ...
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60 views

$f(x+n)$ converges for any $x$ when $n\to +\infty$, then $f(x)$ converges when $x\to +\infty$?

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. I have two questions as follows. Suppose that $f(x+n)\to L$ for all $x\in \mathbb{R}$ when $n\to \infty$. Is it guaranteed that $f(x) \to L$ ...
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Baire's Category Theorem : Open subset of Complate metric space

Question: X is complete metric space and A is non-empty open subset of X then A has to be second category. I'm trying to prove this question and I did the following proof but I'm not so sure. Do I ...
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66 views

Why are these NO counterexamples for Baire's Theorem?

Baire's Theorem states: Let $X$ be a complete metric space and let $U_n \subset X$ be open and dense ($n \in \mathbb{N}$). Then $\bigcap_{n=1}^\infty U_n\subset X$ is dense. In the lectures, we ...
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30 views

Polish group actions: if an orbit is non-meager in itself, it is a Baire space?

Assume that $G$ is a Polish group continuously acting on a Polish space $X$. Let $x \in X$ be a point such that $G \cdot x$, the orbit of $x$, is non-meager in its relative topology. I would like to ...
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60 views

Proof that a particular subset of $L^2[\Pi]$ is dense and first category set (Baire's category)

I hope I don't make too many mistakes since this is my first post. I was trying to prove that $$M = \left[ f\in L^2[\Pi] : \sum_{n=-\infty}^{+\infty}\hat f(n) \quad converges \right]$$ is dense and ...
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The frontier of a set

I'm trying to prove that a certain set is dense in a metric space. I have a metric space X, an open subspace $Y\subset X$ (same metric) and an open set $U\subset Y$ s.t $U$ is open and dense in $Y$. ...
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Proposition 8.23, Kechris (Classical Descriptive Set Theory) meager $F_\sigma$ set contains $A\Delta U$.

Let $X$ be a topological space and $A\subseteq X$. Then the following statements are equivalent: I) $A$ has the Baire property (BP); II) $A=G\cup M$, where $G$ is $G_\delta$ and $M$ is meager; III) $A=...
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Discussion questions involving Liouville numbers

I'm working through Measure and Category by Oxtoby with some friends, but since the book doesn't have any exercises, we needed to come up with our own to discuss. My friend found the following ...
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109 views

Concluding the proof of the Baire category theorem

Let $(M,d)$ be a complete metric space, $A_n\subseteq M$ be nowhere dense for $n\in\mathbb N_0$, $x_0\in M$ and $r_0>0$. We can show that there is a $(x_n)_{n\in\mathbb N}\subseteq M$ and $(r_n)_{...
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48 views

Use Baire’s theorem to show that the set of transitive configurations is dense in $X$.

Let $X :=$ {$0, 1$}$^\mathbb N$. Let $x = x_1x_2 · · · ∈ X$ be a configuration and $w = w_1w_2 · · · w_n ∈$ {$0, 1$}$^∗$ be a word. We say that $w$ occurs in $x$ at position $k$ if $x_kx_{k+1}· · · x_{...
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1answer
50 views

$f$ continuous in a dense set $G_\delta$

If $f:X \to Y$ is a function between a metric space baire $X$ and a metric separable space $Y$ and if for each open $U$ of $Y$, $f^{-1}(U)$ is countable union of closed sets of $X$, then there exists ...
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Munkres section 48 exercise 13 - continuous function space in the fine topology is a Baire space

The following is an exercise in Munkres topology. Let $X$ be a topological space; let $Y$ be a complete metric space. Show that $C(X, Y)$ is a Baire space in the fine topology. [$Hint$: Given basis ...
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35 views

Rudin Real and Complex Analysis - Baire's Category Theorem

I have been trying to understand Rudin's proof of Baire's Category theorem and I think I understood most of it except eqn (2), where he mentions $0 \leq r_1 \leq 1$, and later generalizes this to $0 \...
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57 views

Existence of isolated point.

Suppose: A is a countable closed subset of a complete metric space. Prove: A contains at least one isolated point. (the Baire category theorem brings me no further than Int(A) (the interior of A) is ...
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An extension of Baire's category theorem

In a topological space, a set is said to be rare if its closure has empty interior, and a set is said to be meager if it is a countable union of rare sets. If meager sets all have empty interior, then ...
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What is the category of I in I with Euclidean metric? I is the set of irrationals.

From some topology worksheets I found online that I am using to self-study: Let $I$ be the set of irrationals, and $d$ be the Euclidean metric. What is the category of $I$ in $(I, d)$? I'm sure it is ...
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51 views

Decomposing a general complete seperable metric space into a meager set and a null set

Suppose $X$ is Polish, i.e, a seperable and completely metrizable space. And let $\mu$ be a Borel probability measure on $X$. Furthermore suppose $\mu[\{x\}] = 0$ for every isolated point $x \in X$. ...
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Meagre sets in measured Baire spaces

I know that Baire's category theorem is sometimes used to state results such as "almost all continuous functions on an interval are not differentiable at all points". To say that almost all ...
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107 views

Existence of $L^1((0,1))$ functions which blow up on every open interval

Consider an open interval $(0,1) \subset \mathbb{R}$ and the subset $$ \mathcal{F} := \{f \in L^1((0,1)): \|{f\vert_{(a,b)}}\|_{\infty} = \infty \, \forall \, 0 \leq a < b < 1\} \subset L((0,1), ...
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64 views

How to prove this set is uncountable?

How to prove that the complement in $\mathbb{R}$ of the set $A= \bigcup_{n \in \mathbb{N}} A_n$, where $A_n= (q_n-1/2^n,q_n+1/2^n)$ and $\{q_n\}_{n\in\mathbb{N}} = \mathbb{Q}$, is uncountable ? How to ...
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53 views

Is the intersection of two dense Baire subsets dense?

Let $Y$ be a compact metric space and $Z_1$ and $Z_2$ two dense subsets of $Y$ which are Baire spaces. Is $Z_1\cap Z_2$ dense in $Y$? The answer is obviously yes if $Z_1$ and $Z_2$ are dense $G_{\...
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26 views

Difficulty showing a dense $G_{\delta}$ subset of a Baire space is Baire

I have been working on showing that the irrationals is a Baire space. So far I have shown that the irrationals can be expressed as a $G_{\delta}$ set and I know that if this set was to be Baire then ...
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35 views

A $G_{\delta}$ subset of a Baire space is Baire

I have been seeing this fact used a lot but have not been able to find a proper proof justifying it. Would anyone be able to outline one?
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30 views

Prove that a set does not have Baire property

A set is called meager if it can be written as a countable union of nowhere dense sets. A set $S$ is said to have Baire property if there is an open set $O$ such that the symmetric difference $S\Delta ...
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1answer
49 views

Construct $X$ so that $X$ is not meager and for any non-empty open set $O$, $O\setminus X$ is not meager

A set is called meager is it can written as a countable union of nowhere dense sets. A set $S$ is said to have the Baire property if for some open set $O$, the symmetric difference $S\Delta O$ is ...
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47 views

An application of the Baire category theorem

In the highlighted sentence, $ K_n$ is a closed subset of $V^o$, which is the polar of $V$ and so is compact in the weak-* topology by the Banach-Alaoglu theorem. Therefore, $K_n$ is weak-* compact. ...
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1answer
43 views

Meager set from randomly signing a convergent series

Given a convergent series of real numbers $\sum_{n}a_n$, consider the set $$X=\left\{\sigma:\Bbb N\to \{-1,1\}\mid \mbox{$\sum$}_n \sigma(n)\cdot a_n\text{ converges}\right\}$$ as a subset of the ...
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98 views

The Sorgenfrey plane and the Niemytzki plane are Baire spaces

A space $X$ is called a Baire space if every countable intersection of open dense sets is dense. By the Baire category theorem, every complete metric space is Baire and every locally compact ...
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19 views

Open mapping thm linear spaces Lemma

Extract from Lemma 4.21 M.Einsiedler & T.Ward. $X$ is a normed space, $Y$ is Banach and $T:X \to Y$ is a bounded surjective linear map. The implication of the lemma shows that for any $\epsilon &...
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72 views

Real Analysis Books

I was trying to solve the problem: Is $[0,1]$ a countable disjoint union of closed sets? I find a theorem which is very interesting: Theorem (Sierpiński). If a continuum $X$ has a countable cover $\{...
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32 views

On Čech-complete space.

I'm reading an article of topology and i came across a Properties : Properties : Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete Every Čech-complete space is a ...
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41 views

Proving $F_{\sigma}$ set

I am reading a paper on "On some properties of Baire-$1$ functions". In Proposition $8$, it states that it is straight forward to prove that $E_n$ is an $F_{\sigma}$ set. I would like to ...
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24 views

Baire 1 functions are closed under uniform convergence

May I know what does it mean by "functions are closed" under uniform convergence? Also, how do I then prove that Baire-1 functions are closed under uniform convergence using the epsilon-...
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2answers
66 views

About the proof of Baire's theorem

In my book, the proof of Baire's Theorem starts with: Let $(X,d)$ be a complete metric space. Suppose, by contradiction, that $X$ is of first category, namely that $X=\bigcup_n C_n \,$ where $\, C_n$ ...
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98 views

Subset of a first category set (Baire's category)

I'm trying to prove that any subset of a first category set is of first category. I would like to know if my proof is correct and if there are easier way to prove it. Any check is thankfully ...
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1answer
52 views

sigma-algebra of first category sets and their complements

Let $(X,d)$ a complete metric space and set $$ \mathcal{M} = \{B \subset X : \quad \text{$B$ is of first category $\,$ or $\,$ $B^c=X \backslash B$ is of first category}\} $$ (i) $\quad$ Prove that $\...
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Problem on Baire's category theorem.

I'd like to show that the set of irrational numbers in $[0,1]$ cannot be represented as a countable union of closed sets. The hint says to use Baire's category theorem. I know two versions of such ...
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52 views

subset of $(0,\infty)$ is dense

Let $\Omega \subset (0,\infty)$ an unbounded open set. Consider the set $\Sigma = \{x \in (0,\infty): nx \in \Omega$ for infinitely many $n\}$ Show that $\Sigma$ is dense in $(0, \infty)$. I tried to ...
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46 views

Does there exist a second category set that is not a Baire space?

Can someone give an example of a second-category set $Y$ in a metric space $X$ but $Y$ is not a Baire space. We know that Baire space$\implies $ Second Category. But I am trying to show that the ...
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proof verification: the set of irrational numbers is a dense subset of $\mathbb{R}$

I am trying to prove the proposition that The set of irrational numbers is a dense subset of $\mathbb{R}$ by using the Baire Category Theorem(referred as BCT from now on) Let $\{E_n\}_{n=1}^{\infty}...
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46 views

Set of discontinuity of a derivative.

I was finding a proof of the following result which is a characterisation of functions which are derivative of some function i.e. posseses an antiderivative. Theorem Let $I$ be an interval. $f:\mathbb ...
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49 views

Theorem 2.9 Rudin functional analysis - Inferring exists $n$ such that $K \cap nE \neq \emptyset$

Follow up to this question. I realized that question, which I've asked, explains "why" we can apply Baire's Theorem to $K$. It doesn't address however why $\exists n$ such that $K \cap nE \...
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12 views

Local integrability of differentiable functions

Let $I\subseteq \mathbb{R}$ be an open interval of the real line. Suppose that $f:I\to\mathbb{R}$ is continuous and that the derivative $f'$ of $f$ (meant as the limit of the difference quotient) ...
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1answer
31 views

Can an almost injective quotient map remove interior?

Let $X$ be a second countable compact Hausdorff space without isolated points and let $q: X\to Y$ be a quotient map onto another compact Hausdorff space. Suppose that $X$ has a dense $G_{\delta}$ ...
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90 views

Which spaces have uncountable perfect sets?

I have been thinking about the following question: for which topological spaces $X$ are all perfect subspaces of $X$ uncountable, where perfect means closed with no isolated points. As long as $X$ is $...
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44 views

Singleton in a complete metric space is a complete metric space, but has no interior point. Baire?

I am confused by the above statement with the follwing version of Baire's category theorem: If a non-empty complete metric space $(M,d)$ is the countable union of closed sets, then one of these ...
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1answer
93 views

Baire's theorem: category and density for complements of first category sets

I'm working with the following version of Baire's category theorem: If a non-empty complete metric space $(M,d)$ is the countable union of closed sets, then one of these closed sets has non-empty ...

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