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 $\sf ZF$).

603 questions
11
votes
5 answers

Is the union of two nowhere dense sets nowhere dense?

Is the union of two nowhere dense sets nowhere dense? Using the following definition: A nowhere dense set is a subset $E\subset X$ of a metric space (or topological space) $X$ such that $(\overline{E})^o=\emptyset$. I tried using topological…
Gaston Burrull
  • 5,305
  • 3
  • 28
  • 72
10
votes
0 answers

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO copy: Intersections of open sets and…
Martin Sleziak
  • 50,316
  • 18
  • 169
  • 342
10
votes
1 answer

For a differentiable function $f$ show that $\{x:\limsup_{y\to x}|f'(y)|<\infty\} $ is open and dense in $\mathbb R$

As the title says, given a differentiable function $f: \ \mathbb R \to \mathbb R$ define $$E=\{x:\limsup_{y\to x}|f'(y)|<\infty\} $$ and show that $E $ is open and dense in $\mathbb R$. Here $\limsup_{y\to x}|f'(y)| $ is defined as $\lim_{\epsilon…
MrFranzén
  • 896
  • 1
  • 7
  • 16
10
votes
2 answers

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let $\varepsilon>0$ and $f\in C[0,1]$ be Hölder. Since each Hölder…
9
votes
1 answer

A locally connected metric space of first category

I wonder if there exists a locally connected metric space of first category. I proved a theorem which assumes a space to be connected locally connected hereditarily Baire and metrizable. I specifically wanted to avoid mixing locally connected with…
9
votes
2 answers

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
user107723
  • 467
  • 6
  • 13
9
votes
1 answer

Can a meager linear subspace be written as a countable increasing union of nowhere dense subspaces?

Let $X$ be a separable Banach space. In this question, "subspace" means a linear subspace, not necessarily closed. Suppose $E \subset X$ is a subspace which is meager, so that we can write $E = \bigcup_n E_n$, where the $E_n$ are nowhere dense…
9
votes
2 answers

'Amount' of nowhere-differentiable functions in $C([0,1])$?

A well-known consequence of the Baire Category Theorem that the set of nowhere-differentiable continuous functions is dense in $C([0,1])$. This is often cited as 'almost all continuous functions are nowhere differentiable' (see here), but to me this…
9
votes
1 answer

Show a C-infinity function is a polynomial

Suppose $f\in C^\infty(\mathbb{R})$ and for any $x\in\mathbb{R}$, there exists $N\in\mathbb{N}$ such that $f^{(N)}(x)=0$. Show that $f$ is a polynomial. This is from one of the Analysis qualifying exam problems. I can show there exists an interval…
Chen Ke
  • 489
  • 2
  • 11
8
votes
1 answer

Which spaces have uncountable perfect sets?

I have been thinking about the following question: for which topological spaces $X$ are all perfect subspaces of $X$ uncountable, where perfect means closed with no isolated points. As long as $X$ is $T_1$, we know that perfect sets are at least…
8
votes
0 answers

Elementary example of Baire spaces whose product is not Baire

It is known that there are Baire spaces $X$ and $Y$ whose product is not Baire, the simplest construction I know is due to Cohen and goes as follow: Let $S$ be a stationary subset of $\omega_1$, then forcing with the poset $\Bbb P_S$ of countable…
Alessandro Codenotti
  • 11,165
  • 2
  • 26
  • 51
8
votes
0 answers

Measure of complement of union of nowhere dense set with positive measure

The original question is: Let $A$ be Lebesgue measurable in $\mathbb{R}$ with positive measure. Show that it is not true that there must exist a sequence $\{x_n\}^\infty_{n=1}$ such that the complement of $\bigcup^\infty_{n=1}(A+\{x_n\})$ in…
8
votes
2 answers

Is there a Baire Category Theorem for Complete Topological Vector Spaces?

Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space. Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?
8
votes
1 answer

Help understanding game version of Baire category theorem

I got this from Thomson et al.'s freely available "Elementary Real Analysis" p.356. They introduce Baire's category theorem through a game where, given two players (A) and (B) Player (A) is given a subset $A$ of $\mathbb{R}$, and player (B) is…
JasonMond
  • 3,714
  • 6
  • 34
  • 43
8
votes
1 answer

Determine if the set of all eventually zero sequences is $G_\delta $ in $\ell_2$

For the normed space $\left(\ell^ 2 , \|\cdot\|_2\right) $ I need to check if the following set is of type $ G_\delta $ , $F_\sigma$ : $$E = \left\{ x \in \ell^2 : \mbox{ there exists } N \ge1 \mbox{ s.t } x_n = 0 \mbox{ for each } n \ge N …
user335501
1
2
3
40 41