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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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 ...
4
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0answers
54 views
Partition of $\mathbb{R}$ into nullset and 1st category set
Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$.
Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
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1answer
57 views
Use the concept of Baire Category to prove that the upper integral of a positive function is always positive.
Use Baire Category to prove that the upper integral from $0$ to $1$ of any $f:[0,1]\to(0,1]$ is greater than $0$.
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0answers
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Baire's theorem [duplicate]
I want to know interesting applications of Baire's Category Theorem.
For example existence of no where differentiable function.
Can any body tell me some similar applications?
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1answer
31 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 ...
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2answers
54 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 ...
4
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2answers
98 views
About Baire's Category Theorem(BCT)
Consider the following theorem known as Baire's Category Theorem (BCT).
Theorem.[BCT] A non-empty complete metric space $X$ is not a countable union of nowhere dense sets.
I am interested on how to ...
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2answers
91 views
Cantor's intersection theorem and Baire Category Theorem
From an old post in math stackexchange, I read a comment which goes as follows " I like to think of Baire Category Theorem as spiced up version of Cantor's Intersection Theorem". My question -----is ...
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3answers
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Dirichlet Function Pointwise Convergence
We say that a function $f$ is Baire Class $1$ if there is a sequence of functions $f_i \to f$ pointwise where each $f_i$ is continuous.
The set of discontinuities of a Baire Class $1$ function $f$ ...
0
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2answers
119 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 ...
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1answer
86 views
The Principle of Condensation of Singularities
Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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2answers
130 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.
6
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1answer
308 views
Infinitely times differentiable function
Let $f$ belong to $ C^{\infty}[0,1]$ and for each $x \in [0,1]$ there exists $n \in \mathbb{N}$ so that
$f^{(n)}(x)=0$. Prove that $f$ is a polynomial in $[0,1]$.
I am trying to use Baire Category ...
2
votes
1answer
101 views
Baire: Show that $f\colon \mathbb{R}\to\mathbb{R}$ is a polynomial in an open bounded set
Let $f\colon\mathbb{R}\to\mathbb{R}$ be an infinitely differentiable function, and suppose that for each $x\in\mathbb{R},$ $\exists\ n=n(x)\in\mathbb{N}$ such that $f^{(n)}(x)=0$. For each fixed ...
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1answer
175 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
202 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 ...
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1answer
417 views
Let $X$ be an infinite dimensional Banach space. Prove that every basis of X is uncountable.
Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable.
Can anyone help how can I solve the above problem?
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2answers
159 views
A question on the proof of Open mapping theorem
I was following the proof of the Open Mapping Theorem in functional analysis in Wikipedia, and I came across a line in the proof that did not make sense.
Some notations: $U,V$ are open unit balls in ...
5
votes
1answer
100 views
Is $\ell^1 \subset \ell^2$ meagre? [duplicate]
Possible Duplicate:
Prove $\ell_1$ is first category in $\ell_2$
Consider $\ell^2$ with the topology induced by the usual norm. We can easily prove that $\ell^1 \subset \ell^2$. I am ...
4
votes
1answer
115 views
Prove $\ell_1$ is first category in $\ell_2$
Prove that $\ell_1$ is first category in $\ell_2$.
I tried to solve this, but had no idea about the approach. Any suggestions are helpful.
Thanks in advance.
5
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1answer
75 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 ...
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1answer
105 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.
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3
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1answer
176 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
107 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 ...
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4answers
200 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 $\mathbb{R}^k$.
...
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2answers
95 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):
...
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16answers
2k 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 ...
84
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1answer
2k 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 ...
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1answer
105 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 ...
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2answers
164 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
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1answer
186 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
109 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 ...
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2answers
485 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 ...
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2answers
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Second category but not locally residual
Let $X$ be a Baire space.
A subset $E\subset X$ is said to be of first category if it can be expressed as the union of countably many nowhere dense subsets of $X$. Then $E$ is said to be of second ...
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1answer
252 views
Every space is “almost” Baire?
There is this theorem called the Banach category theorem which states that in every topological space any union of open sets of first category is of first category.
Now doesn't this imply that every ...
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0answers
60 views
x-section of closure of E of first category implies x-section of E nowhere dense
Let $E$ be a subset of first category of product space $X \times Y$.
Why is the following true: if $(\bar E)_x \subset Y$ is of first category then it follows that $E_x$ is nowhere dense.
$E_x$ ...
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2answers
325 views
Doubt in Kechris's Classical Descriptive Set Theory
In Theorem 8.29 in the Kechris's book Classical Descriptive Set Theory, he writes that if $W=\bigcup_{i\in I} U_i$ where $U_i$ are pairwise disjoint and set $A$ is comeager in each $U_i$, then $A$ is ...
6
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1answer
152 views
Help understanding game version of Baire category theorem
I got this from Thomson et al.'s freely available "Elementary Real Analysis" p.356.
They introduce Baire's category theorem through a game where, given two players (A) and (B)
Player (A) is given ...
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3answers
469 views
Corollary to Baire's Category Theorem
In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem:
"In a complete metric space, the intersection of any countable ...
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5answers
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Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
33
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5answers
3k 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?
2
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1answer
153 views
disjoint union of Baire spaces which is a Baire space
Say we have a family {$A_\alpha$} of disjoint Baire spaces. Also suppose that each $A_\alpha$ is disjoint from the closure of the union of the other sets. Show that $\bigcup_{\alpha} A_\alpha$ is a ...
