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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Is the set of $\varepsilon$-discontinuities closed if it is defined without $\limsup$?
Let $f_n \in C([0,1])$ be a sequence of functions with $f_n(x) \to f(x)$ for all $x$. Consider the set of discontiuities of $f$, which I denote by $\mathcal{D}$. Then, we can define:
$$\mathcal{D} = \...
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Metric set with empty interior, Baire's theorem
I'm solving the following problem:
Let $(M,d)$ be a complete metric space, with $M=\cup_{n=1}^\infty F_n$, where $F_n$ is closed. Show that there is an $F_n$ such that $int(F_n) \neq \emptyset$
My ...
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Are left-inverses of continuous functions pointwise limits of continuous functions?
Let $f : X \to Y$ be a surjective continuous map and suppose $g: Y \to X$ satisfies $g \circ f = \text{id}_X$. If we have that $Y$ is compact and $X$ and $Y$ are both subsets of $\mathbb{R}^n$, does ...
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Variant of Baire theorem
I consider $(X,d)$ a complete metric space. I have this weak form of the Baire theorem : There does not exist nonempty open subset $O$ of $X$ such that $O=\bigcup_{n\geq 0} F_n$ where the $F_n$ are ...
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What's wrong with this "counter-example" of Baire's Theorem?
Yesterday my teacher had taught us the Baire Space, that is, a topological space where the intersection of countably many dense open subsets is always dense.
I immediately came up with an example of ...
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Density of smooth functions whose derivatives have analytical/closed form expression
I was thinking about the following question the other day and thought I'd ask here to see if anyone can make it more precise and/or make any mathematical sense: What is the density (in terms of ...
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Inverse limits of complete metric spaces is Baire
It is well known that arbitrary products of complete metric spaces are Baire (refer to Dugundji, example).
But, what happens when one considers inverse limits of complete metric spaces over an ...
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How is the set $C(f)\cap V$ of second category in $V$?
I am reading the paper "P. S. Kenderov, I. S. Kortezov and W. B. Moors, Continuity points of quasi-continuous mappings, Topology Appl. 109 (2001), 321–346." Just before Theorem 2 of the ...
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Compact subsets of Hausdorff $h$-measure $0$ has complement of first category
We work on the space $\mathcal{K}$ of compact subsets of $[0,1]$, with the Hausdorff metric.
Let $h:[0,1]\to \mathbb R$ be a continuous strictly increasing function with $h(0)=0$. We say that a ...
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Pointwise limit of functions on $[0,1]$
i was thinking on the problem bellow but I couldn't fully solve the problem, here is the statement:
Let $f:[0,1] \rightarrow [0,1]$ be a continuous function such that $\forall x \in [0,1]$ there ...
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If $f$ is a continuous and non negative function such that $\sum_{n=1}^\infty f(na)$ converges $\forall a\ge0$, prove that the convergence is uniform
Let $f: [0,+\infty) \to \mathbb{R}^+$ a continuous and non negative function, $f(x) \ge 0$, so that $\sum_{n=1}^\infty f(na)$ converges $\forall a \ge 0$. I would like to prove that $\sum_{n=1}^\infty ...
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Graph of a monotone function is nowhere dense.
Graph of a monotone real function defined on an interval is a nowhere dense set in $\mathbb{R}^{2}$.
I know that when $f$ is continuous, $G(f)$ is a closed set. Also, its interior is empty since any $...
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