Tagged Questions

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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Let $X$ be an infinite dimensional Banach space. Prove that every Hamel 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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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 ...
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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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A classical problem about limit of continuous function at infinity and its connection with Baire Category Theorem

When I google "baire category theorem", I get a link to Ben Green's website. And at the end of the paper, he mentioned such a classic problem: Suppose that $f:\mathbb{R}^+\to\mathbb{R}^+$ is a ...
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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 $n\... 5answers 2k views Why is it that$\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}$is not a complete metric space. But completeness is not a topological invariant, so why is the ... 4answers 435 views (ZF) If$\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$. ... 1answer 197 views Prove$\ell_1$is first category in$\ell_2$Prove that$\ell_1$is first category in$\ell_2$. I tried to solve this, but had no idea about the approach. Any suggestions are helpful. Thanks in advance. 1answer 901 views Diophantine number has full measure but is meager This an exercise 3 on Terence Tao's blog: A real number$x$is Diophantine if for every$\varepsilon > 0$there exists$c_\varepsilon > 0$such that$|x - \frac{a}{q}| \geq \frac{c_\...
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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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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 ...
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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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Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
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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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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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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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Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty

The following is problem 3.22 from Rudin's Princples of Mathematical Analysis: Suppose $X$ is a nonempty complete metric space, and $\{G_n\}$ is a sequence of dense open subsets of $X$. Prove ...
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Infinite intersection between a arbitrary set of integers and a set of floor powers

Let $E$ be an infinite set of positive integers, proves that there is a $\alpha \in \mathbb{R}$ such that $\{\left \lfloor \alpha^k \right \rfloor ;k \in \mathbb{N} \}\cap E$ is infinite. I have two ...
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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 ...
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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 no ...
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Meager subset of $2^\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
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Is $c_0$ meagre in $l^\infty$?

I have been reading up on the meagre sets sans an instructor after a 10 year break from this sort of mathematics and I would like to test my understanding. I want to know if the following is true (and ...
Borel set as union of $G_\delta$ and countable set
What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?
Use Baire Category to prove that the upper integral from $0$ to $1$ of any $f:[0,1]\to(0,1]$ is greater than $0$.