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.

The Baire category theorem (BCT) asserts that a countable intersection of open dense sets is still dense. A topological space for which the BCT holds is called a Baire space. Examples include locally compact spaces and complete metric spaces. The BCT has major applications in real and functional analysis, such as the open mapping theorem, or the existence of many nowhere differentiable functions.

A meager set is a set which is a countable union nowhere dense sets. They are also called sets of first category. Non-meager sets are called sets of second category. An equivalent formulation of the BCT is that all open subsets of a topological space $X$ are sets of second category.

The theorem is provable from the Zermelo–Fraenkel set theory with the axiom of choice, $\sf ZFC$, but not from $\sf ZF$ (in fact BCT is equivalent to the Principle of Dependent Choice, or $\sf DC$, over $\sf ZF$).

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Why need open in the Baire Category Theorem

In the statement of the Baire Category Theorem, one needs to include openness of a set so that the theorem holds. Question: what is the example such that countable intersection of dense sets is not dense? EDIT: perhaps I should rephrase my…
Idonknow
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Corollary of Baire theorem

Can anybody help me to prove the following result? Corollary of Baire theorem: Let $(K_j)_{j>0}$ be an increasing sequence of compact sets in $C^n$ and $X$ a bounded open set such that $\overline X\subset\cup K_j.$ Then for every $x\in X$ there is…
Kara
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On the zero set of a $C^2$ function on $[0,1]^2$

Let $f:[0,1]^2\rightarrow \mathbb{R}$ be a twice continuously differentiable function with the property that for all $x\in [0,1]$, there is an interval $I_x\subset [0,1]$ such that $f(x,y)=0$ for all $y\in I_x$. Does it follow that there must exist…
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Is $\{1/n : n = 1, 2, 3, ...\}$ completely metrizable?

Is $\{1/n : n = 1, 2, 3, ...\}$ with the subspace topology from $\mathbb{R}$ completely metrizable? As as result of Baire's category theorem, we know that if a metric space is complete and there are no isolated points, then the space is…
mr eyeglasses
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Locally varying, continuous functions on $R^2$, show $R^2$ cannot be written as $\cup^\infty_{i=1}\cup^\infty_{j=1} \{x : f_i(x) = c_j\}$\}

Problem Statement: If a real valued function on $\mathbb R^2$ is locally varying (on any non-empty open subset $U \subset \mathbb R^2$, the function is not constant), show that $\mathbb R^2$ cannot be written as $$\mathbb R^2 =…
poppy3345
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Is there an example of a topological space that is of the Second Baire Category but is not a Baire space?

By being of the second category I mean that it is not the countable union of nowhere dense sets and by Baire space I mean a space such that a countable intersection of open dense sets is dense in X. It's easy to see that if it is a Baire space then…
Jack
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How is Baire category theorem used here?

The following is a doubt that arouse from reading this paper by Bandyopadhyay, Jarosz and Rao. Let $E$ be a Banach space and $E^{*}$ be its dual space. Let $e_{0}$ be an element of norm one in $E$ such that its associated state space…
Arundhathi
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Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
adoion
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intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that $A\bigcap xB$ is of the second category,…
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Baire category theorem in a Banach space

For any two distinct $u,v$ in a countable dense subset of separable real Banach space $X$, let $S(u,v) = \{f \in Y \mid f(u)=f(v)\}$, where $Y$ is the dual space of $X$. Each of $S(u,v)$ is a proper closed subspace of $Y$. How can I prove that the…
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Application of the Baire category theory

Definition: A set $M\subset X$ is called "of first category" if it is countable union of nowhere dense sets. Otherwise its called "of second category". I want to see whether the following sets are of first or second category: 1) $\mathbb Q$ 2)…
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Question regarding the proof that every hamel basis of an infinite space is uncountable

I am reading the following question: Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable. And I am wondering why $$X=\bigcup_{n\in \mathbb N}X_n$$ Since the right hand side is just an union of some sets…
Duke
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Two proofs with possibly Baire category theorem about completness.

I'm working with completness right now and I've come across two interesting problems. In my opinion they are worth a little bit attention . a) Let $K$ be closed subset with empty interior on euclidean line. Prove that $\exists \ {t \in \mathbb{R}}$…
MatJ
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Baire category theorem in use on a plane

Let $F\subset\mathbb{R}$ be a closed nowhere dense set. One must show there exists $(a,b)\in S^1$ for which $b\neq qa+c$, for all $q\in\mathbb{Q},c\in F$. It's my second question concerning Baire category theorem - I already know what the crux is --…
Jules
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Prove that $M$ is $\sigma$-algebra (Baire categories)

Let $(X,d)$ be a complete metric space and set $$ M := \{B \subset X : B \text{ is of first category or the complement of a first category set}\} $$ Prove that $M$ is a $\sigma$-algebra. In particular, I'm having issues dimostrating that: $$\forall…
Julian Vené
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