# Questions tagged [baire-category]

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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### Prove that $[0,1]$ is not a disjoint union of a countable family of closed sets [closed]

This proof is probably too long, please don't downvote this. I'm just leaving it here in case somebody wants to actually verify its validity for me. I didn't use a hint in the book and decided to ...
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### Nowhere Dense Set and Corollary to Baire's Theorem

Not Homework just Personal Study Reference text: Kolmogorov and Fomin - Introductory Real Analysis I am working on understanding why a complete metric space with no isolated points is uncountable. The ...
1 vote
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### Is there a metric $d$ such that $(\mathbb{F}[x], d)$ is complete?

Denote $\mathbb{F}[x]$ (where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$) is the space consisted of polynomials with coefficients in the field $\mathbb{F}$. The question is whether there ...
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### How can I show that $X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$ is not complete?

Let $X$ be a vector space of real polynomials, $$X=\left\{\sum_{k=0}^d a_kt^k:~d\geq 0,a_k\in \Bbb{R}\right\}$$I want to show that $X$ is not compleate with respect to any norm on $X$. My idea was to ...
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### Verify: The points of discontinuity of any $f: \mathbb R \to \mathbb R$ are a $F_\sigma$ set

Prove that the points of discontinuity of an arbitrary function $\mathbb R \to \mathbb R$ are a $F_\sigma$ set. Note: Proofs are available; this question is to verify and critique this proof and ...
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### Verification of the converse of Baire Category Theorem

One way to state the Baire Category Theorem is as follows: If $X \neq \emptyset$ is a complete metric space, then $X$ is nonmeager as a subset of itself. Then it was mentioned that the converse, that ...
1 vote
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### Category of continuous functions with compact support in the space of continuous bounded functions and in itself

All functions here are from R to R and I use the supremum metric d. $C_c$= continuous functions with compact support and $C_b$ = continuous bounded functions. We say a metric space $(X,d)$ is of first ...
1 vote
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### Does there exist a nonmeager compact subset of an infinite-dimensional Banach space?

Let $X$ be an infinite-dimensional Banach space and let $K \subset X$ be compact. If $K$ is contained in a finite dimensional subspace of $X$, then $K$ must be meager. However, there may ...
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### If $B$ and $C$ are residual sets in $\mathbb{R}$, then $B-C = \mathbb{R}$

The definition of a "residual set" I'm using is "complement of a meager set", and a meager set is a countable union of nowhere dense sets. I've thought about approaching this in a ...
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### Is the set $A$ nowhere dense?

A part of an exercise I've been trying to solve asks to prove that the set $$A = \left\{ \sum_{n=0}^{\infty} a_n x^n \, \Big| \, (a_n) \in \ell^1(\mathbb{N}) \right\} \subset C([0,1])$$ is not ...
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### a question between the first category set and closed proper subspace

Assume $X$ is a complete norm space with $\mathbb{dim}~X=+\infty$, $E\subset X$ is a proper closed subspace. On one hand, I have known $E$ is a nowhere dense set, because any proper closed subspace of ...
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### Countable Intersection of Sets of Second Category

We say a set is of first category if it is a union of countably many nowhere dense subsets. If a set is not of first category, we say it is of second category. Now I am thinking of a question. Suppose ...
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### If $f(nc) \to 0$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$ [duplicate]

Suppose $f$ is a continuous function on $\Bbb R$. If $f(nc) \to 0$ as $n \to \infty$ for all $c>0$ then prove that $f(x) \to 0$ as $x \to \infty$ I have solved the problem after extending the ...