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.

learn more… | top users | synonyms

0
votes
1answer
25 views

Closed set of 1st category

If $F$ is a closed subset of a complete metric space, is it possible for $F$ to be of the first category? This seems to lead to a contradiction. Since $F$ is a closed subset of a complete metric ...
3
votes
2answers
3k views

Definitions of Baire first and second category sets

From Planetmath A meager or Baire first category set in a topological space is one which is a countable union of nowhere dense sets. A Baire second category set is one which contains a ...
0
votes
1answer
34 views

Show that C^α([0, 1]) is of first category in C([0, 1]).

Recall that for any $0 < α < 1,$ the space $C^\alpha ([0, 1])$ is the set of continuous functions on $[0, 1]$ with norm of f = sup |f| + $ sup \ x\ne y$ |f(x) − f(y)|/|x − y|^α< ∞, equipped ...
2
votes
1answer
47 views

Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense.

Let $f: \mathbb R \rightarrow \mathbb R$ be a continuous function that is nonconstant on any interval. Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense. My ...
1
vote
1answer
111 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 ...
0
votes
2answers
40 views

Why need open in the Baire Category Theorem

In the statement of the Baire Category Theorem, one needs to include openness of a set so that the theorem holds. Question: what is the example such that countable intersection of dense sets is not ...
3
votes
1answer
47 views

Baire category of subset $\mathbb{R} \setminus B$ with lebesgue zero measure $B$

Let $\mathbb{Q}=\{ q_i\}^{\infty}_{i=1}$. Let $I_{ij} = \left(q_i - \delta_{ij}, q_i + \delta_{ij}\right)$, $\delta_{ij}=2^{-i-j-1}$ for all $i,j \geq 1$ be an open interval. For every $j\geq1$, ...
2
votes
1answer
67 views

Baire's Theorem and Irrationals

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
4
votes
0answers
62 views

Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...
0
votes
1answer
67 views

Corollary of Baire theorem

Can anybody help me to prove the following result? Corollary of Baire theorem: Let $(K_j)_{j>0}$ be an increasing sequence of compact sets in $C^n$ and $X$ a bounded open set such that $\overline ...
1
vote
1answer
48 views

On the zero set of a $C^2$ function on $[0,1]^2$ [closed]

Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I_x\subset [0,1]$ such that $f(x,y)=0$ for ...
1
vote
2answers
47 views

Proving a result in Baire categories

If $A\subset X$ is a $G_\delta$ set and dense in $X$, then what I want to show is that $X\setminus A$ is of the first Baire category. Where the meaning of first category is that I can write ...
4
votes
2answers
149 views

Equivalence of Baire Space definitions

I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent. Defn1: Any topological space X such that the intersection of any countable ...
1
vote
1answer
56 views

continuous an open image of a meager set is meager

I want to know if the following is true. Let $X$ and $Y$ be topological spaces and $f\colon X\to Y$ a continuous open surjection. Suppose that $X$ is meager, then $Y$ is meager. Recall that a meager ...
4
votes
1answer
53 views

Question related to Baire category theorem

I met this question from a faculty booklet which I had trouble on involving metric spaces, stating: Let $ (X,d) $ be a complete metric space with the sequence of closed sets, $ \{F_n\}_{n \in N} $ ...
0
votes
1answer
55 views

Is $\{1/n : n = 1, 2, 3, …\}$ completely metrizable?

Is $\{1/n : n = 1, 2, 3, ...\}$ with the subspace topology from $\mathbb{R}$ completely metrizable? As as result of Baire's category theorem, we know that if a metric space is complete and there ...
177
votes
1answer
6k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
0
votes
1answer
15 views

Locally varying, continuous functions on $R^2$, show $R^2$ cannot be written as $\cup^\infty_{i=1}\cup^\infty_{j=1} \{x : f_i(x) = c_j\}$\}

Problem Statement: If a real valued function on $\mathbb R^2$ is locally varying (on any non-empty open subset $U \subset \mathbb R^2$, the function is not constant), show that $\mathbb R^2$ cannot be ...
0
votes
1answer
23 views

Is there an example of a topological space that is of the Second Baire Category but is not a Baire space?

By being of the second category I mean that it is not the countable union of nowhere dense sets and by Baire space I mean a space such that a countable intersection of open dense sets is dense in X. ...
0
votes
1answer
34 views

How does this follow from the Baire category theorem?

The book says that statement 2 is a direct consequence of statement 1. I don't see how they prove statement 2 directly from statement 1, can you please help me? Statement 1: A complete metric ...
3
votes
1answer
48 views

Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials

Problem statement: Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials. Note that the zero set of a polynomial $p(x,y)$ is $\{(x,y) : p(x,y) = 0$}. My ...
0
votes
1answer
34 views

How is Baire category theorem used here?

The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ ...
1
vote
0answers
35 views

Lebesgue characterization of Baire class 1 functions

Lebesgue's characterization of Baire class one functions on $\mathbb R$ is the following: $f:\mathbb R \rightarrow \mathbb R$ is Baire class one iff for all $r\in \mathbb R$ $\{ x: f > r \}$ and ...
2
votes
1answer
92 views

Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
1
vote
1answer
46 views

Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...
1
vote
1answer
230 views

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 ...
2
votes
1answer
78 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 ...
5
votes
1answer
224 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 $A_n$ ...
1
vote
1answer
100 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) ...
3
votes
2answers
156 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 ...
2
votes
2answers
56 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) ...
0
votes
1answer
94 views

a question on Pixley-Roy topology

Let $X$ be a $T_1$ space and let $F[X]$ be $\{x\subset X:\text{is finite}\}$ with Pixley-Roy topology. If $X$ is not discrete, how to prove $F[X]$ is not a Baire space? Thanks ahead:) Definition ...
1
vote
1answer
53 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 ...
0
votes
0answers
41 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
0
votes
0answers
66 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 ...
1
vote
1answer
109 views

Baire's Theorem Formulation

I am still having problems with some of the proofs of Baire's theorem. In Introductory Real Analysis by Kolmogorov and Fomin The statement of the theorem states that: "A complete metric space R ...
7
votes
2answers
212 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 ...
0
votes
1answer
134 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this? I guess $A$ must be an $G_{\delta}$ set which is dense in $\Bbb ...
3
votes
1answer
45 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 ...
1
vote
1answer
72 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 $A$, ...
0
votes
2answers
81 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 ...
0
votes
1answer
46 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 ...
2
votes
1answer
88 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 ...
1
vote
1answer
67 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 ...
1
vote
0answers
46 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 ...
3
votes
1answer
44 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 ...
1
vote
1answer
40 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 ...
3
votes
1answer
70 views

Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
0
votes
1answer
68 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 ...
59
votes
6answers
7k views

Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?