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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Definitions of Baire first and second category sets

From Planetmath A meager or Baire first category set in a topological space is one which is a countable union of nowhere dense sets. A Baire second category set is one which contains a countable union of open and dense sets. From Wikipedia: A…
Tim
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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 infinitely-dimensional F-space has a countable…
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An interesting problem involving recursion

Given a continuous function $f:[0, 1] \rightarrow [0, 1]$. Here we denote $f^n(x) = f(f^{n-1}(x))$. For every $x_0 \in [0, 1]$ there exists $n \in \mathbb{N}$ such that $f^n(x_0) = 0$. Prove that $f^N(x) \equiv 0$ for some $N$. I've only managed to…
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In an uncountable Polish space there is no countably generated $\sigma$-algebra between analytic sets and sets with the Baire property

Let $X$ be an uncountable Polish space. Assume that $\mathcal{A}$ is a $\sigma$-algebra of subsets of $X$ such that every analytic subset of $X$ belongs to $\mathcal{A}$ and every member of $\mathcal{A}$ has the Baire property. Then $\mathcal{A}$ is…
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Existence of $L^1((0,1))$ functions which blow up on every open interval

Consider an open interval $(0,1) \subset \mathbb{R}$ and the subset $$ \mathcal{F} := \{f \in L^1((0,1)): \|{f\vert_{(a,b)}}\|_{\infty} = \infty \, \forall \, 0 \leq a < b < 1\} \subset L((0,1), dx). $$ I want to show that $\mathcal{F}$ is non-empty…
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Example of a Baire metric space which is not completely metrizable

I know that some Baire metric spaces are not complete metric spaces but all examples, that I know, are completely metrizable. Help me to find an example of Baire metric space which is not completely metrizable. $[$Please give some short proofs or…
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Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup |\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in [0,1)$

From Gamelin and Greene's Introduction to Topology, 2nd edition, chapter 1 section 6 (Continuity): Show that there is a continuous real function $h$ on $[0,1]$ such that $\lim \sup _{t \to 0^+}|\frac{h(x+t)-h(x)}{t}| = \infty$ for all $x \in…
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The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j \in\Bbb N\} =+\infty$. Then there is an $x$ (indeed a…
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Are absolutely convergent series "many" or "few" compared to conditionally convergent series?

We can identify absolutely convergent series with the $l^1$ space and conditionally convergent series with a subspace of $c_0 = \{\{a_n\} \in l^\infty : a_n \to 0\}$ Since $l^1$ contains all finite sequences, it is dense in $c_0$, so in this sense…
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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 Baire's theorem, namely, that $\bigcap_{n=1}^\infty G_n$…
crf
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Existence of nowhere differentiable functions

I have troubles with one step of my solution to this problem: Show that there exists a continuous function $f:[0,1]\to \mathbb{R}$ which is not differentiable at any point. Hint: Consider $X=C([0,1],\mathbb{R})$ and $$U_n:=\left\{f\in…
user185346
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Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, Trans. Amer. Math. Soc., 281(1984), no. 1, 209--213"…
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Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?

Does there exist a function $f:\mathbb R \to (0,\infty)$ such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$ ?
user228169
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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 usually, we start with fomulating two sets $$Q =…
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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 topological space X (let's say which is not of…
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