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 proof

> A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets. I don't understand the following point the proof: we assume that $X$ is the countable union of ...
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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?
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1answer
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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 ...
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1answer
134 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 ...
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definition of Baire first category set

Suppose that $A$ is of the first category. Is it possible to write $A$ as a countable union of sets that are not necessarily nowhere dense? I mean after all a first category set is defined such that ...
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1answer
113 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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1answer
157 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 ...
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0answers
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Baire Category Theorem with $G_{\delta}$

If we have a subset $Y$ of a complete metric space $X$ such that $Y$ is a $G_{\delta}$, then how would I go to show that $Y$ satisfies the Baire category theorem? I am trying to get there by showing ...
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1answer
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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 ...
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3answers
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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 ...
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1answer
211 views

Proving the existence of a non-monotone continuous function defined on $[0,1]$

Let $(I_n)_{n \in \mathbb N}$ be the sequence of intervals of $[0,1]$ with rational endpoints, and for every $n \in \mathbb N~$ let $E_n=\{f \in C[0,1] : f \:\text{is monotone in}\: I_n\}$. Prove that ...
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1answer
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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 ...
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1answer
68 views

Genericity and category

This paper by Ambos-Spies and Mayordomo on the theory of algorithmic randomness introduces the notion of genericity saying that it is based on Baire category while the usual notion of randomness is ...
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0answers
36 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 ...
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1answer
105 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 ...
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2answers
31 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
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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 ...
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1answer
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Measure spaces are proper subsets

I want to prove that $L^2$ is of the first category in $L^1$, thus I have to prove that $L^2$ is the countable union of nowhere dense subsets. The hint I get is: Take $g_n(x)=n$ for $0\leq ...
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1answer
99 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 ...
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1answer
32 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 ...
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1answer
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$X$ 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 ...
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1answer
277 views

Diophantine number has full measure but is meager

This an exercise 3 on Terence Tao's blog: A real number $x$ is Diophantine if for every $\varepsilon > 0$ there exists $c_\varepsilon > 0$ such that $|x - \frac{a}{q}| \geq ...
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2answers
177 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 ...
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1answer
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$X$ 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.
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1answer
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Can a sequence of functions have infinity as limit exactly at rationals?

Someone asked me this question. And he said it's an exercise from Rudin's Real and Complex Analysis. Does there exist a sequence of continuous functions $f_n(x)$, such that $\lim_{n \to \infty} ...
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1answer
239 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 ...
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1answer
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Slight generalisation of the Baire category theorem?

By the Baire category theorem, one cannot write a complete metric space $X$ as a countable union of closed nowhere dense subsets of $X$. Can this be generalised to say that there is no injection $f: X ...
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1answer
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Why is there an element with infinite spectrum in a commutative Banach algebra with infinitely many characters?

Let $A$ be a commutative Banach algebra such that set of all characters is infinite. I want to prove that there exists an element in $A$ such that its spectrum is infinite.
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2answers
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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 ...
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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
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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
133 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
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$\mathbb{Q}$ 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 ...
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2answers
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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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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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1answer
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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
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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
340 views

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 $f$ ...
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255 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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2answers
443 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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2answers
300 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.
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1answer
333 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 ...
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1answer
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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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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 ...
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1answer
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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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1answer
153 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.
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1answer
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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 ...
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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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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. $E_x$ ...