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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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The only meagre and open set is the empty set?

Suppose that A is a meagre set in a topological space X. I think that the answer of my question is yes, because $\mathring {A}$ is empty. Am I right?
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Baire's Theorem: Examples for open dense subsets

Theorem (Baire): Let $(X,d)$ be a complete metric space and $(D_n)_{n \in \mathbb{N}}$ a family of open dense subsets of $X$. Then $\bigcap_{n \in \mathbb{N}} D_n$ is also dense in $X$. This is the ...
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Isolated points in a complete metric space

I need to prove that if $X$ is a countable (infinite) complete metric space, then $X$ has infinitely many isolated points. I have read that the Category Baire theorem implies that X should have at ...
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Baire's theorem and compact sets [closed]

Prove or disprove the following statement. There is a countable collection of compact sets $\{K_n\}$ of $C([0,1])$ so that $K_n \subseteq K_{n+1}$ for all $n$ and that $$C([0,1]) = \bigcup_{n=1}^\...
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Proof of Lemma preceding Principle of Condensation of Singularities

Under the Wikipedia page for the Principle of Uniform Boundedness, we have the Corollaries of the Uniform Boundedness Principle. The third of these relates to the Principle of Condensation of ...
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Proving that $\left[ 0,1 \right] $ is not the countable union of disjoint closed intervals using Baire Category Theorem

I already showed that that set of all end points of the intervals are closed. Letting $E$ be this set, we know that $(E,d)$ is a complete metric space. Ultimately I want to use the Baire category ...
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A multivariable complex analysis problem probably related with Baire Category Theorem

The original problem is an extension of the Hurwitz theorem into the case of multivariables, which states that given a region $D\in \mathbb{C}^n$, if the sequence of analytical functions $f_k(z),z\in ...
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If for some sequence $a_n\to \infty$ the limit $\lim_{n\to\infty} f(a_nx)$ exists for all $x\in\mathbb R$, then $\lim_{x\to\infty} f(x)$ exists

A little bit of context. I was given a problem which went like if $X_n$ is normally distributed with mean $a_n$ and is converging in distribution to $X$, then $a_n\to a$ for some $a\in\mathbb R$ and $...
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1answer
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How to prove Baire Category theorem by Zorn's lemma?

Baire Category theory states that, in a complete metric space, the union of countably many dense open sets is dense. The proof relies on Axiom of choice. But AC is equivalent to Zorn's Lemma. So can ...
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Can we find $V_n$ in such a way that $\bigcap\limits_{n=1}^{\infty} V_n$ is countable?

If $V_n$ is open, dense subset of $\mathbb R$ for each $n\in \mathbb N$ then by Baire's Category theorem we know that $\displaystyle \cap_{n=1}^{\infty} V_n$ is dense in $\mathbb R$. My question ...
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Are there any vector spaces that cannot be given a norm that makes the vector space a complete metric space?

And if so, how can one prove that there is no such norm? I suppose one can use the form of Baire Category Theorem which states that a complete metric space cannot be written as a countable union of ...
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Compact Hausdorf space $\implies$ not the countable union of nowhere dense sets? [duplicate]

I can sort of see this intuitively, seeing as it's a similar argument to the Baire Category theorem, but does anyone have a proof I could look at? Would it suffice to say that every locally compact ...
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Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets

I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it. The Baire Category theorem asserts that if X is a complete metric space ...
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Does the intersection of two dense, open sets have the Baire property?

Definition: $A \subseteq X$ for a metric space $X$ has the Baire Property if for any sequence of sets {$V_{n}$} for $n \geq 1$ that are dense and open in $A$, $$cl(\cap V_{n}) \cap A = A $$ for all $...
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Is the Set of Continuous Functions that are the Sum of Even and Odd Functions Meager?

Consider $X = \mathcal{C}([−1,1])$ with the usual norm $\|f\|_{\infty} = \sup_{t\in [−1,1]}|f(t)|.$ Define $$\mathcal{A}_{+}=\{ f \in X : f(t)=f(−t) \space \forall t\in [−1,1]\},$$ $$\mathcal{...
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Baire's Category Theorem Proof

I've searched online for this specific direction of proving Baire's Category Theorem but couldn't find anything substantive. I believe this attempts to prove it in the contrapositive way that, for ...
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Suppose $X$ is a complete metric space. Is the complement of a countable union of nowhere-dense sets a dense set?

I am able to prove this if we have a countable collection $\{E_n\}$ of closed nowhere-dense subsets, however I am being asked to prove it for a countable collection of nowhere-dense sets that need not ...
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1answer
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Equivalence of Baire Category Theorem for Complete Metric Spaces.

In my studies of functional analysis I have come across a statement of the Baire Category Theorem which states: "The Baire Category Theorem says that if a complete metric space $X$ is a ...
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$X^*$ with $weak^*$-topology is first category in itself when $X$ is infinite dimensional Fréchet space

I am reading Rudin's Functional Analysis and stuck at the problem in exercise 11 on page 87 : Let $X$ be infinite dimensional Fréchet space , then $X^*$ with it's $weak^*$-topology is of first ...
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Problem 2. A comprehensive course in Analysis. Barry simon. Page 239.

Definition (Baire set) Let X be a compact Hausdorff space. The Baire sets are the smallest $\sigma$-algebra containing all compacts $G_{\delta}$'s. Definition (Partition) Given an algebra , $\...
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For an infinite dimensional Banach space, $X^*$ when given the weak* topology is of the first category in itself [duplicate]

Let $X$ be an infinite dimensional Banach space. Why is $X^*$ of the first category in itself when given the weak* topology. Very closely related to $X^*$ with its weak*-topology is of the first ...
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Problem 4 Barry Simon a comprehensive course in analysis part 1.

(a) For any bounded Baire function, $f$, on a compact Hausdorff space $X$, prove that $||f||_{\infty}:=\inf\left\{\sup_x |g(x)|: f-g=0\text{ for a.e. } x\right\}$ exists, defines a seminorm, and ...
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Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

This is a follow-up to my question here. The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be ...
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Proving that the set of continuous nowhere differentiable functions is dense using Baire's Category Theorem

I'm trying to prove the next problem: Let $C([0,1],\mathbb{R})$ the space of continuous function $f:[0,1]\to \mathbb{R}$ with the supremum(uniform convergence) metric and let $\mathbb{B}\subset C([0,...
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1answer
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On Borel subsets of a topological space

This is exercise from Megginson's Banach Space Theory. I have to show that if $B$ is a Borel subset of a topological space $X$, then there exist sets $M, N$ of first category in $X$ such that $(B\...
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An application of Baire's theorem

Good evening, let $1_\mathbb{Q}$ be the characteristic function of the rational numbers, $(f_n)_{n\in \mathbb{N}}$ a sequence with $f_n\in C([0,1],\mathbb{R})$ and $$\lim_{n\to\infty}f_n(x)=1_{\...
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1answer
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Why a set of second category minus (take away) a set of first category is of second category?

Why if $A$ is a set of second category and $B$ is a set of first category, then $A \setminus B$ is of second category? (I am a native Spanish speaker, so that forgive me if my English is not perfect) ...
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Are the two formulations of Baire category theorem equivalent for arbitrary metric spaces?

As we all know, Baire Category theorem has two equivalent forms $X$ is a complete metric space, then the countable intersection of dense open sets is nonempty. $X$ is a complete metric space, $X$ is ...
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Zorn's Lemma in an Oxtoby's theorem

I'm reading a theorem due to Oxtoby about the link between the Baire spaces and the Choquet Game. I found it in Kechris, Classical Descriptive Set Theory, pag. 43. I can't understand what he means ...
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Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup |\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in [0,1)$

From Gamelin and Greene's Introduction to Topology, 2nd edition, chapter 1 section 6 (Continuity): Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup _{t \to 0^+}|\...
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Is a nowhere dense set closed?

I am trying to prove an equivalent weak version of Baire's category theorem which states that $:$ Every complete metric space is a set of second category or non-meagre. I am trying to ...
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Negation of countable union of nowhere dense.

According to my references, a topological space is said to be of first category if it can be expressed as countable union of nowhere dense set, where a set is said to be nowhere dense if it's closure ...
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$[0,1]$ is uncountable using Baire's Category Theorem

Given that, $\mathbb{R}$ is complete. $[0,1] \subset \mathbb{R}$ and is closed $\implies [0,1]$ is complete and clearly non-empty. So we know by BCT: $[0,1]$ is second category. Suppose $[0,1]$ ...
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Show that the set of $f\in L^{1}$ so that $f\notin L^{p}$ is residual.

Consider $L^{p}([0,1])$ with Lebesgue measure. Note that if $f\in L^{p}$ with $p>1$ then $f\in L^{1}$. Show that the set of $f\in L^{1}$ so that $f\notin L^{p}$ is residual. First of all, a set $E\...
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Show that a closed subset is first category.

I have the following problem: Suppose $A$ is a closed subset of a complete metric space $X$. Show that $A$ is of the first category if and only if $A$ has empty interior. My attemp: if $A$ is a ...
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Countable Dense Set not a $G_{\delta}$ set? Where's the contradiction?

Prove the following theorem: Theorem: If $D$ is a countable dense subset of $\Bbb{R}$, there is no function $f : \Bbb{R} \to \Bbb{R}$ that is continuous precisely at the points of $D$. (a) ...
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Understanding a step in Baire Category Theorem

I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is ...
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Two variants of proof for Baire's category theorem

One of the variants of the so called Baire's Category Theorem (BCT) says that Given a (possibly nonempty) complete metric space $(X,d)$ and a system of open dense subsets $(O_n)$ of $X,$ then $\...
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Subspace $\ell^2$ is of first category

Let $\ell^2$ be the Hilbert space of square summable sequences, and $\mathcal{H}$ be the subspace consisting of sequences $\{x_n\}$ with $\sum_{n=1}^\infty n^2|x_n|^2<\infty$ Show that $\mathcal{...
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Baire category and the existence of very continuous functions

I want to prove the existence of a $f:C([0,1])\to \mathbb{R}$ such that $$\lim_{h\to0}\sup|h|^{-\alpha}|f(x+h)-f(x)|\to \infty$$ for all $x\in [0,1]$. Where $\alpha > 0$. I was given the hint that ...
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Baire Category and Decimal Expansion

I am trying to improve my understanding and application of the Baire Category theorem and got stuck on the following problem; Let $A$ be the set of real numbers whose decimal expansion does not use ...
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Baire category theorem and lower semicontinuous functions

Let $(X, \tau)$ be a Baire space, $I$ an index set and for each $x \in X$, let the set $\{f_i(x) : i \in I\}$ be bounded above, where each mapping $f_i : (X,\tau) \to \mathbb{R}$ is lower ...
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On the classic exercise “$f\in C^\infty(\mathbb{R})$ s.t. $\forall x \exists n\in\mathbb{N}: f^{(n)}(x)=0$ implies $f$ is a polynomial”

The title is pretty much self-explanatory; Problem: Let $f\in C^\infty(\mathbb{R})$ such that for each $x\in\mathbb{R}$ there exists $n_x\in\mathbb{N}$ s.t. $f^{(n_x)}(x)=0$. Prove that $f$ is a ...
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Space of $L^p$ functions not in $L^q$ for all $q\neq p$

Fix $p\geq 1$, and consider the set $S$ of all $f\in L^p(\mathbb R)$ with the property that $f\not\in L^q(\mathbb R)$ whenever $1\leq q\leq\infty$ and $q\neq p$. For example, the function $f$ defined ...
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Proof of Uniform boundedness principle (why are the sets closed)

I have a question on the proof of the UBP: Let $(X,d)$ be a complete metric space and $\mathcal{F}$ be a familly of continuous functions such that for each $x \in X$ $\sup_{f \in \mathcal{F}}|f(x)|...
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A question on the proof of the Baire Category theorem

In many proofs (but not all) of the Baire Category theorem one requires on the n:th induction step that: $\overline{B(y_n,r_n)} \subset U_n \cap B(y_{n-1},r_{n-1})$ (where $\{ U_n \}_{n \geq}$ is a ...
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If $f \in C^\infty (\mathbb{R})$ such that $f^{(n)}=0$ for some $n$, then $f$ is a polynomial?

Just as in the title, suppose $f \in C^\infty ((a,b),\mathbb{R})$ such that $f^{(n)}=0$ for some $n\in\mathbb{N}$. Prove that $f$ is a polynomial. My solution to this is to use the Taylor expansion ...
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Applications of Baire Category Theorem to Connected Sets

There are some wonderful and well known applications of the Baire Category theorem that prove the existence of certain objects without constructing them, e.g. there exists a continuous nowhere ...
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On the notion of Cech-complete

I read this old question, and in particular I was interested in this answer. Ostap says that for the concept of Cech-complete a generalization of the Baire's theorem holds, but he didn't say that a ...
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Uniqueness in Baire property representation for compact Hausdorff spaces

Let $X$ be a compact Hausdorff space. I know that every Borel set $B$ is congruent to a regular closed set $R$ modulo a meager set $M$. In other words, $B\oplus R=M$ (where $\oplus$ denotes the ...