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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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A lemma of Cielsielski

Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (...
sabrina's user avatar
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1 answer
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$\mathbb{N}$ is uncountable?

I recently saw a proof that $\mathbb{R}$ is uncountable using Baire's Category Theorem. It goes like this: Suppose $\mathbb{R}$ is countable then $\mathbb{R} = \cup(x_n)$. Since $\mathbb{R}$ is ...
Jackson Smith's user avatar
1 vote
1 answer
33 views

Prove equivalent form of Baire's Category Theorem

I'm trying to prove these two statements of Baire's Category Theorem are equivalent: Let X complete metric space. A subset of X is meagre if it can be written as the countable union of nowhere dense ...
Jackson Smith's user avatar
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Is the set of $\varepsilon$-discontinuities closed if it is defined without $\limsup$?

Let $f_n \in C([0,1])$ be a sequence of functions with $f_n(x) \to f(x)$ for all $x$. Consider the set of discontiuities of $f$, which I denote by $\mathcal{D}$. Then, we can define: $$\mathcal{D} = \...
J. S.'s user avatar
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1 answer
37 views

Metric set with empty interior, Baire's theorem

I'm solving the following problem: Let $(M,d)$ be a complete metric space, with $M=\cup_{n=1}^\infty F_n$, where $F_n$ is closed. Show that there is an $F_n$ such that $int(F_n) \neq \emptyset$ My ...
nileebolt's user avatar
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4 votes
1 answer
80 views

Are left-inverses of continuous functions pointwise limits of continuous functions?

Let $f : X \to Y$ be a surjective continuous map and suppose $g: Y \to X$ satisfies $g \circ f = \text{id}_X$. If we have that $Y$ is compact and $X$ and $Y$ are both subsets of $\mathbb{R}^n$, does ...
J. S.'s user avatar
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Variant of Baire theorem

I consider $(X,d)$ a complete metric space. I have this weak form of the Baire theorem : There does not exist nonempty open subset $O$ of $X$ such that $O=\bigcup_{n\geq 0} F_n$ where the $F_n$ are ...
G2MWF's user avatar
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1 answer
143 views

What's wrong with this "counter-example" of Baire's Theorem?

Yesterday my teacher had taught us the Baire Space, that is, a topological space where the intersection of countably many dense open subsets is always dense. I immediately came up with an example of ...
yummy's user avatar
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1 answer
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Density of smooth functions whose derivatives have analytical/closed form expression

I was thinking about the following question the other day and thought I'd ask here to see if anyone can make it more precise and/or make any mathematical sense: What is the density (in terms of ...
Dowdow's user avatar
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Inverse limits of complete metric spaces is Baire

It is well known that arbitrary products of complete metric spaces are Baire (refer to Dugundji, example). But, what happens when one considers inverse limits of complete metric spaces over an ...
mathable's user avatar
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2 votes
1 answer
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How is the set $C(f)\cap V$ of second category in $V$?

I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the ...
Ghosh Da's user avatar
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0 answers
10 views

Compact subsets of Hausdorff $h$-measure $0$ has complement of first category

We work on the space $\mathcal{K}$ of compact subsets of $[0,1]$, with the Hausdorff metric. Let $h:[0,1]\to \mathbb R$ be a continuous strictly increasing function with $h(0)=0$. We say that a ...
ThetaOmega's user avatar
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62 views

Pointwise limit of functions on $[0,1]$

i was thinking on the problem bellow but I couldn't fully solve the problem, here is the statement: Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $\forall x \in [0,1]$ there ...
Amir Mg's user avatar
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If $f$ is a continuous and non negative function such that $\sum_{n=1}^\infty f(na)$ converges $\forall a\ge0$, prove that the convergence is uniform

Let $f: [0,+\infty) \to \mathbb{R}^+$ a continuous and non negative function, $f(x) \ge 0$, so that $\sum_{n=1}^\infty f(na)$ converges $\forall a \ge 0$. I would like to prove that $\sum_{n=1}^\infty ...
Carloss's user avatar
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2 votes
1 answer
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Graph of a monotone function is nowhere dense.

Graph of a monotone real function defined on an interval is a nowhere dense set in $\mathbb{R}^{2}$. I know that when $f$ is continuous, $G(f)$ is a closed set. Also, its interior is empty since any $...
huh's user avatar
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1 answer
124 views

Must a subset of the real line that is comprised entirely of condensation points be a Baire space?

Let $X$ be a subset of the real line in which every point is a condensation point. Is $X$ a Baire space?
Kiran Antony's user avatar
1 vote
0 answers
42 views

Exercise about repeated primitives

Suppose ${f}_{n}$ is a sequence of $C([0,1])$, ${f'}_{n+1}={f}_{n}$ for all $n$ and: $\forall x \in [0,1]$, $\exists n$ : ${f}_{n}(x)=0$. I want to show that ${f}_{1}$ is identically null! So I think ...
Albi's user avatar
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2 answers
152 views

$\lim\limits_{n\rightarrow\infty}f(2^{n}x)=0$ implies$\lim\limits_{x\rightarrow\infty}f(x)=0?$

If $f: (0,\infty)\rightarrow\mathbb{R}^{1}$ is continuous, and $\lim\limits_{n\rightarrow\infty}f(2^{n}x)=0.$ for all $x>0$. I want to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$, but I have no ...
lkj123's user avatar
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1 vote
1 answer
43 views

Ergodic transformations form a $G_\delta$ set in the weak topology of the automorphism group.

If $(X,\mathcal{L},\mu)$ is a Lebesgue-standar space and $G$ is its group of automorphisms, i know that the set of all ergodic transformations $\mathcal{E}$ is a $G_\delta$ set in the weak topology. ...
Susana Santoyo's user avatar
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Showing a certain dense subset of $\mathcal{C}^1([0,1]^2,\mathbb{R}^4)$ is residual.

For every $f\in X\doteq \mathcal{C}^1([0,1]^2,\mathbb{R}^4)$, define $S_f=\{x\in[0,1]^2:f_1(x)f_2(x)-f_3(x)f_4(x)=0\}$. I am wondering if the set $G=\{f\in X:S_f\text{ is contained in the union of ...
Andrew Murdza's user avatar
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0 answers
48 views

Are the irrationals Gδ because they are uncountable?

https://math.stackexchange.com/a/61110/1248511 I just saw this proof for showing that the rationals are not a Gδ set(without using baire's theorem) and while I'm not really sure about what exactly I'm ...
NoetherBoy 's user avatar
1 vote
1 answer
32 views

Convex lsc function on a Banach space and Baire's theorem

I am struggling with a exercise from the book "Functional Analysis, Calculus of Variation and Optimal Control". The goal is to show the following claim: A convex lower-semi-continuous (lsc) ...
P.Jo's user avatar
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2 votes
1 answer
98 views

Essentially discontinuous on a dense subset of $\mathbb{R}$, but continuous on the complement which is also dense

$A, B$ is a partition of $\mathbb{R}$, and both $A$ and $B$ are dense. Is there a function $f: \mathbb{R} \to \mathbb{R}$, that is essentially discontinuous ($\lim f$ does not exist) at everywhere in $...
Y.D.X.'s user avatar
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0 answers
73 views

Understanding the Proof of the Smale Sard Theorem

I am trying to understand the proof of Theorem 1.3 from https://people.math.osu.edu/burghelea.1/MaterialDiffTopology/Smale.pdf It is stated as follows: Let $f:M\to V$ be a $C^q$ Fredholm map with $q&...
Andrew Murdza's user avatar
2 votes
0 answers
624 views

Prove that $[0,1]$ is not a disjoint union of a countable family of closed sets [closed]

This proof is probably too long, please don't downvote this. I'm just leaving it here in case somebody wants to actually verify its validity for me. I didn't use a hint in the book and decided to ...
Dymista's user avatar
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2 votes
0 answers
42 views

Nowhere Dense Set and Corollary to Baire's Theorem

Not Homework just Personal Study Reference text: Kolmogorov and Fomin - Introductory Real Analysis I am working on understanding why a complete metric space with no isolated points is uncountable. The ...
Mud's user avatar
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1 vote
1 answer
40 views

Is there a metric $d$ such that $(\mathbb{F}[x], d)$ is complete?

Denote $\mathbb{F}[x]$ (where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$) is the space consisted of polynomials with coefficients in the field $\mathbb{F}$. The question is whether there ...
Bakkune's user avatar
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2 votes
0 answers
63 views

f(tn) converges -> f(t) converges [duplicate]

Claim: Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a continous function, for which $\text{lim}_{n \rightarrow \infty}f(nt) = 0$ ($n \in \mathbb{N}$) holds for all $t \in [0, \infty)$. Then $\text{...
Jahi02's user avatar
  • 301
1 vote
1 answer
48 views

If $(X, \tau)$ is a Baire space, then $X \times [0,1]_{std}$ is a Baire space

I’m trying to prove the following claim: “If $(X, \tau)$ is a Baire space, then $X \times [0,1]_{std}$ is a Baire space” The only way I know how to prove this is by using a Theorem from John C Oxtoby, ...
obitobi_tobias's user avatar
7 votes
3 answers
733 views

Why topological spaces in Baire category theorem are required to be Hausdorff?

In Baire category theorem it says a locally compact Hausdorff space $S$ is second category. In the proof, it choose $V_1,V_2\cdots$ are dense open subset of $S$. $B_0$ is an arbitrary nonempty open ...
xyz's user avatar
  • 709
2 votes
0 answers
74 views

Baire Category in Exercise 4 of Chapter 2, Functional Analysis Rudin

In the next problem of Rudin's book, Functional Analysis, Let $L^1$ and $L^3$ be the usual Lebesgue spaces on the unit interval. Prove that $L^2$ is of the first category in $L^1$, in the following ...
Adrian's user avatar
  • 61
4 votes
1 answer
170 views

A corollary of Baire Category Theorem

I have learnt that the theorem says: Let $\{U_n\}_{n=1}^\infty$ be a sequence of dense open subsets of a complete metric space X. Then $\displaystyle\cap_{n=1}^\infty U_n$ is also dense in X. I also ...
Lee Wei Xuan's user avatar
2 votes
0 answers
66 views

Let E be a non-meager subset of a Banach space, then E - E contains a neighborhood of zero.

I want to prove the statement: Let E be a non-meager subset of a Banach space, then E - E contains a neighborhood of zero. Here, E - E means the Minkowski difference of E with itself. The statement is ...
User84308209's user avatar
3 votes
0 answers
99 views

Can a proper subspace of a Banach space be a countable intersection of open sets?

Let X be a Banach space and suppose E is a dense linear subspace which is a Gδ -set. Show that E = X. By the Baire Category theorem we can show that E is non-meager (second category), but I don't know ...
User84308209's user avatar
2 votes
0 answers
75 views

When Should I Try to Use the Baire Category Theorem?

I've studied the statement and proof of the Baire Category Theorem for complete metric spaces and locally compact Hausdorff spaces, and I've used the theorem in an exercise to prove that that in ...
Nick A.'s user avatar
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-2 votes
1 answer
86 views

A exercise assigned in the topology course

Let $\{x_n\}_{n=1}^{\infty}$ be an enumeration of $\Bbb Q$, $\{y_\lambda\}_{\lambda \in \Lambda}$ an enumeration of $\Bbb R\setminus\Bbb Q$. Proof: for any families $\{r_n\}_{n=1}^\infty,\,\{ \rho_\...
anyon's user avatar
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3 votes
1 answer
115 views

Alternative proofs of $L^q[0,1]$ is a (Baire) first category subset in $L^p[0,1]$

Let $1\leq p<q<\infty$, deduce from each one of the following statements that $L^q[0,1]$ is a (Baire) first category subset in $L^p[0,1]$: Let $p<r<s<q$ and $\beta=s/(s-1)$, we define $...
Sergio Ferrer's user avatar
0 votes
1 answer
109 views

Brezis' Exercise 4.8

I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 4.8 Let $(\Omega, \mathcal F, \mu)$ be a $\sigma$-finite measure space. Let $X$ be a closed vector subspace of $L^1 (\...
Akira's user avatar
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2 votes
2 answers
209 views

Nowhere dense subsets of a dense subspace are nowhere dense in the whole space and vice versa

It seems, with the following lemma, the proposition at the bottom easily follows. If $Y\subset X$ dense. Then, for every nonempty $A\subset Y$, $\text{Int}_Y \left(\text{cl}_Y A\right)=Y\cap \text{...
user760's user avatar
  • 1,670
0 votes
1 answer
56 views

Equivalent definitions of Baire spaces

We say that a metric space $X$ is a Baire space if there is no open set $E$ such that $$E \subseteq \bigcup\limits_{n\geq 1} F_i,$$ in which each $F_i$ is a closed set with empty interior. Suppose ...
José Victor Gomes's user avatar
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0 answers
103 views

Examples of dense $G_\delta$ sets in $L^p$ spaces

The Baire Category Theorem states that the countable intersections of open dense subsets of a complete metric space (called dense $G_\delta$ sets) are dense. Any open set is $G_\delta$, so any dense ...
Andrew Murdza's user avatar
1 vote
0 answers
82 views

Baire category theorem and countable union of closed meagre sets in complete metric space

In my functional analysis course Baire category theorem is the following: Complete metric spaces are of second category. The following was stated as Baire Category theorem's consequence: Suppose $X$ ...
H-a-y-K's user avatar
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1 vote
1 answer
56 views

Why is this proof of the Baire Category Theorem (BCT) is incorrect?

The Baire Category Theorem states: Theorem: Let $X$ be a complete metric space. Suppose that each element of the sequence $\{U_n\}$ of open sets of $X$ is dense. It holds that $\cap^\infty U_n$ is ...
Kzards's user avatar
  • 151
-2 votes
1 answer
129 views

the strength of the axiom of choice used in forcing

I believe that Shelah's model gives a ZFC theorem of relative consistency results: $ZFC \vdash Con(ZFC + \text{there is a strongly inaccessible cardinal})\rightarrow Con(ZF+DC+ \text{all sets are ...
clhs's user avatar
  • 29
1 vote
1 answer
60 views

How can I show that $X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$ is not complete?

Let $X$ be a vector space of real polynomials, $$X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$$I want to show that $X$ is not compleate with respect to any norm on $X$. My idea was to ...
user1294729's user avatar
  • 2,018
0 votes
1 answer
194 views

Verify: The points of discontinuity of any $f: \mathbb R \to \mathbb R$ are a $F_\sigma$ set

Prove that the points of discontinuity of an arbitrary function $\mathbb R \to \mathbb R$ are a $F_\sigma$ set. Note: Proofs are available; this question is to verify and critique this proof and ...
SRobertJames's user avatar
  • 4,450
2 votes
1 answer
81 views

Verification of the converse of Baire Category Theorem

One way to state the Baire Category Theorem is as follows: If $X \neq \emptyset$ is a complete metric space, then $X$ is nonmeager as a subset of itself. Then it was mentioned that the converse, that ...
user57's user avatar
  • 796
1 vote
0 answers
48 views

Category of continuous functions with compact support in the space of continuous bounded functions and in itself

All functions here are from R to R and I use the supremum metric d. $C_c$= continuous functions with compact support and $C_b$ = continuous bounded functions. We say a metric space $(X,d)$ is of first ...
autodidacti's user avatar
1 vote
1 answer
72 views

Does there exist a nonmeager compact subset of an infinite-dimensional Banach space?

Let $ X $ be an infinite-dimensional Banach space and let $ K \subset X $ be compact. If $ K $ is contained in a finite dimensional subspace of $ X $, then $ K $ must be meager. However, there may ...
jzs's user avatar
  • 73
0 votes
1 answer
46 views

a question between the first category set and closed proper subspace

Assume $X$ is a complete norm space with $\mathbb{dim}~X=+\infty$, $E\subset X$ is a proper closed subspace. On one hand, I have known $E$ is a nowhere dense set, because any proper closed subspace of ...
fusheng's user avatar
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