4
votes
0answers
182 views

Questions related to intersections of open sets and Baire spaces

I was curious a little about the classes of topological spaces described below. For each of them I want to ask whether they have been studied and whether they have some interesting properties. ...
1
vote
0answers
22 views

If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
1
vote
1answer
41 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
1
vote
1answer
23 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 ...
0
votes
2answers
36 views

Baire space but not locally compact

I need an example that is a Baire space but not locally compact. I think, $\mathbb{R}^ \mathbb{R}$ is such an example. $\mathbb{R}^ \mathbb{R}$ is not locally compact. But I could not proof that it ...
2
votes
3answers
39 views

Meagre Sets: Algebra

Let meagre subsets be defined as: $A\text{ meagre}\iff A=\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ with }\overline{A_k}°=\varnothing$ Then it satisfies: $B\subseteq A\text{ meagre}\Rightarrow ...
1
vote
1answer
30 views

Baire: Equivalent Statements II

Moreover, why does it follow for Baire Spaces and why is it strictly weaker that: $X=\bigcup_{k=1}^\infty A_k\quad\Rightarrow\quad\exists k_0\in\mathbb{N},x_0\in X: A_{k_0}\in\mathcal{N}_{x_0}$ ...
2
votes
2answers
71 views

Baire: Equivalent Statements

How to proof that these statements are equivalent: Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. ...
1
vote
1answer
40 views

Prove that a set is dense and of the first category in $L^2(T)$

Define the Fourier coefficients $\hat{f}(n)$ of a function $f\in L^2(T)$, $T$ is a unit circle, by: $\hat{f}(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(e^{i\theta})e^{-in\theta}d\theta$ Let ...
3
votes
2answers
132 views

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
1
vote
1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
1
vote
1answer
50 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
4
votes
1answer
42 views

Baire Category Theorem proof in Gamelin Greene - how do they shrink the closure of open ball

I am confused by a step in the Gamelin and Greene proof of the Baire Category Theorem. Here is the start of the proof. Theorem: Let $\{U_n\}_{n=1}^{\infty}$ be a sequence of dense open subsets of a ...
0
votes
1answer
60 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
2
votes
1answer
30 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
3answers
78 views

Is a Baire Space necessarily complete?

A complete metric space a Baire space. But is a Baire space necessarily complete?
1
vote
1answer
45 views

Baire Category Theorem: What should we really prove there?

I am reading about the Baire Category Theorem in Jech's book on set theory. 4.8: Baire Category Theorem: Let $D_0,D_1,\dots,D_n,\dots$, $n \in \mathbb{N}$, be open dense subset of $\mathbb{R}$. Then ...
0
votes
1answer
87 views

No infinite-dimensional $F$-space has a countable Hamel basis

If $X$ is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove that $X$ is of the first category in itself. Prove that therefore ...
2
votes
1answer
110 views

Examples of rare, meager and nonmeager sets in $\mathbb{R}$

Kreyszig Functional Analysis book presents the following definition. I'm trying to get some examples. (a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so ...
1
vote
1answer
109 views

Can somebody help me to understand this? (Baire Category Theorem)

Theorem $\mathbf{6.11}$ (Baire Category Theorem) Every residual subset of $\Bbb R$ is dense in $\Bbb R$. $\mathbf{6.4.5}$ Suppose that $\bigcup_{n=1}^\infty A_n$ contains some interval ...
0
votes
1answer
60 views

Baire Category Theorem in a Smooth Manifold

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$. How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using Baire category theorem? and which of the theorem version is ...
0
votes
0answers
33 views

Sets with the property of baire in proof of Poincare Recurrence Theorem

I have a problem with the proof of Thm 17.1 in Oxtoby's Category and Measure. In statement of the Poincare Recurrence Theorem he does not say anything about property of Baire, but in the proof he ...
0
votes
1answer
84 views

Any example of open set of first category?

In the book by Oxtoby, the chapter 16 is devoted to them and Banach Category Theorem. However I do not understand this chapter and parts of the following one. I know that I can define any topology I ...
0
votes
2answers
29 views

Confusion about this assertion

I have seen this result true in general ? Every zero set is $G_\delta$-sets. As I know in Normal spaces, every closed $G_\delta$ set is zero set. Thx in advance
4
votes
1answer
135 views

$L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b) [duplicate]

We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$. But in (b) they ask to ...
1
vote
1answer
91 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
0
votes
1answer
31 views

In a Baire space $X$, if an open set meets a nonmeager set, is the intersection nonmeager?

In a Baire space $X$, if an open set $U$ meets a nonmeager set $N$, is the intersection nonmeager? If they do not meet then it's false; take the upper half line of the reals. It does not meet some of ...
0
votes
1answer
40 views

$X$ a separable metric space with no isolated points. If $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager?

Let $X$ be a separable metric space with no isolated points. Then if $G \subset X$ is a countable dense $G_{\delta}$ subset of $X$, why is $X$ meager? I've written a couple things so far, Let ...
6
votes
2answers
159 views

Examples of closed subspaces of Baire spaces that fail to be Baire?

I am looking for some nice examples of Baire spaces containing closed subspaces that fail to be Baire. Clearly, $X$ should not satisfy either one of the standard hypotheses for the Baire category ...
2
votes
1answer
37 views

$X$ is a complete metric space, $Y$ is compact. $X \times Y$ is Baire?

Requesting a hint or solution. X is a complete metric space and Y is a compact hausdorff space. Trying to show that $X \times Y$ is a Baire space.
2
votes
1answer
227 views

Are function spaces Baire?

Let $X$ and $Y$ be manifolds and suppose that $X$ is a compact, complete metric space and $Y$ is a complete metric space. So, both $X$ and $Y$ are Baire spaces. Question: For what values of $k\geq ...
4
votes
2answers
104 views

Baire's theorem from a point of view of measure theory

According to Baire's theorem, for each countable collection of open dense subsets of $[0,1]$, their intersection $A$ is dense. Are we able to say something about the Lebegue's measure of $A$? Must it ...
2
votes
1answer
114 views

Complement of a meagre subset of $\mathbb{R}$ contains an uncountable $G_\delta$ set

I'm trying to show that the complement of a meagre set $A \subseteq \mathbb{R}$ contains an uncountable $G_\delta$ set. Here is what I got so far : since $A$ is meagre, there exists nowhere dense ...
1
vote
2answers
78 views

$\mathbb{Q}$ is not locally compact using baire category?

is there any result related together with Baire Category Theorem, Locally compactness, and Completeness? actually I would like to prove $\mathbb{Q}$ is not locally compact. I realize that singleton ...
0
votes
2answers
213 views

Sets of second category-topology

A set is of first category if it is the union of nowhere dense sets and otherwise it is of second category. How can we prove that irrational numbers are of second category and the rationals are of of ...
2
votes
2answers
262 views

Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.
0
votes
1answer
483 views

Is the union of two nowhere dense sets nowhere dense?

Is the union of two nowhere dense sets nowhere dense? Using the following definition: A nowhere dense set is a subset $E\subset X$ of a metric space (or topological space) $X$ such that ...
2
votes
2answers
427 views

Baby Rudin problem 3.22: prove Baire's theorem. Am I going in a reasonable direction?

Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, namely, that $\bigcap_1^\infty G_n$ is not empty. Hint: find a ...
5
votes
1answer
82 views

A property dealing with complete metric spaces

I came across a property in a textbook that caught my eye. The property is: If $X$ is a complete metric space, then the intersection of any two dense $G_{\delta}$-subsets of $X$ is dense in $X$. This ...
1
vote
1answer
146 views

(ZF) Equivalent statements to Baire Category Theorem

So far, I have proved following two for a polish space $X$; 1.If $\{F_n\}$ is a family of closed subset of $X$, where $X=\bigcup_{n\in \omega} F_n$, then at least one $F_n$ has a nonempty inteior. ...
3
votes
1answer
248 views

Baire Category Theorem

This is Asaf's argument; (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior Suppose that $(X,d)$ is a separable complete metric space, and ...
5
votes
4answers
142 views

Complement of co-dense set.

Asaf's argument : (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior Let $X$ be a separable complete metric space. Let $D$ be a countable debse ...
4
votes
4answers
281 views

(ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior

Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF. Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$. ...
2
votes
2answers
156 views

$x$-axis is meager set on $\mathbb{R}^2$

Subset $A$ of metric space $X$ is meager on $X$, iff $\text{IntCl}A=\emptyset$. But, why $x$-axis is meager set on $\mathbb{R}^2$? My attempt (please don't kill me): ...
63
votes
18answers
6k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
3
votes
1answer
146 views

Question missing condition in Royden Exercise 7.42 b, about Baire Category

In Royden's Real Analysis P164 Q7.42b, It assumes that $X$ and $Y$ are complete metric spaces. Let $O$ be a dense open set in $X \times Y$. Assertion: Then there is a $G \subset X$ which is a ...
3
votes
2answers
229 views

Example of Baire Space

Can anybody supply an example of a Baire Space, that is neither locally compact nor metrizable. I would be gratefull also for some references.
4
votes
1answer
280 views

Application of Baire category theorem in Moore plane

The proof that Moore plane is not normal I have read was using Cantor's nesting theorem. But I heard that it is also possible to use Baire category theorem to prove and I want to know how. So, as ...
3
votes
1answer
133 views

Baire Category and F sigmas

Can anyone give an example of a Meagre subset of $\mathbb{R}$ (i.e., of the first Baire Category), which isn't an $F_\sigma$? Simple cardinality arguments show that such a thing exists, but I can't ...
0
votes
2answers
1k 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 ...