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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A lemma of Cielsielski

Reading section 2 of this paper '' A century of Sierpinski–Zygmund functions'' (Ciesielski, K.C., Seoane-Sepúlveda, J.B.. Rev. R. Acad.Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113(4), 3863–3901 (...
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$\mathbb{N}$ is uncountable?

I recently saw a proof that $\mathbb{R}$ is uncountable using Baire's Category Theorem. It goes like this: Suppose $\mathbb{R}$ is countable then $\mathbb{R} = \cup(x_n)$. Since $\mathbb{R}$ is ...
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Prove equivalent form of Baire's Category Theorem

I'm trying to prove these two statements of Baire's Category Theorem are equivalent: Let X complete metric space. A subset of X is meagre if it can be written as the countable union of nowhere dense ...
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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 ...
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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 ...
• 401
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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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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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