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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Every perfect Polish space is a continuous 1-1 image of the Baire space

I am learning descriptive set theory by myself and stumble upon this theorem without proof, could somebody help me prove it: Every perfect Polish space is a continuous 1-1 image of the Baire space ...
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28 views

On the proof of Baire category theorem

I would like to ask about the proof of Baire Category theorem found on wolfram. The excerpt is as below: Baire's category theorem, also known as Baire's theorem and the category theorem, is a result ...
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Producing $\mathbb{R}$ with countable amount of sets?

Prove, that you can't "produce" $\mathbb{R}$ with countable amount of sets, which are nowhere dense(I am not sure I said this definition correct, with nowhere dense, I mean that $Int(\overline X) = ...
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81 views

If $f:\mathbb R \to \mathbb R$ is an additive function whose graph is $G_{\delta}$ in $\mathbb R^2$ , then the graph is closed in $\mathbb R^2$?

If $f:\mathbb R \to \mathbb R$ is an additive function i.e. $f(x+y)=f(x)+f(y) ,\forall x,y \in \mathbb R $ such that $G(f):\{(x,f(x)) : x\in \mathbb R\}$ is a countable intersection of open sets , ...
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1answer
23 views

Is it true that every 1st category subset of a 2nd category space has empty interior?

Let $X$ be a metric space. Are these conditions equivalent: Each set of the 1. category in $X$ has empty interior; $X$ is of the 2. category. It is obvious that $1 \Rightarrow 2$. Is it true that ...
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1answer
24 views

Why empty set? (consequences of Baire's theorem)

I did not understand the proof of Theorem 5.13 of Rudin, [Real and Complex Analysis]. See next. In a complete metric space X which has no isolated points, no countable dense set is a $G_{\delta}$. ...
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Proving a linear function is bounded using the Baire category theorem (or its consequences)

This is a problem from Folland. Let $\mathcal{X}, \mathcal{Y}$ be Banach spaces. If $T : \mathcal{X} \rightarrow \mathcal{Y}$ is linear and $f \circ T \in \mathcal{X}^*$ for all $f \in ...
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1answer
23 views

Second category subset vs subspace

Let $(X,T)$ be a topological space. When we say $A$ is first category in $(X,T)$ we mean that it is the union of (or is covered by) countably many sets which are nowhere dense in $(X,T)$. $A$ is ...
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41 views

Non differentiable solutions to $\partial_x f + \partial_y f =0$

This nice paper by Gilles Godefroy (in French) tells us the story of Baire's lemma. In 1896, Monsieur Baire was lecturing on analysis and carelessly gave the following exercise: to find all solutions ...
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1answer
32 views

Implication of Baire Category Theorem

The statement that the countable intersection of open dense sets is dense is equivalent to the statement that the countable union of nowhere dense sets contains no balls. When proving the ...
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1answer
39 views

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 ...
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83 views

Compact Hausdorff space is of second category

Let $X$ be a compact Hausdorff space, prove $X$ is of second category. I found a proof of this theorem in the case of locally compact Hausdorff spaces. Let $E_n$ be open and dense in $X$, locally ...
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1answer
97 views

Differentiable function with a set of critical points of second category.

I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words: Let $U \subset \mathbb{R}$ open. Is there a differentiable ...
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1answer
24 views

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 $$X=\bigcup_{n\in \mathbb N}X_n$$ ...
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1answer
78 views

If $X$ is complete, then there is no continuous and open $\,f:X \to \mathbb{Q}$

I've encountered the following question and got stuck : There is no continuous and open mapping $\,f:X \to \mathbb{Q},$ where $X$ is a complete metric space. I thought it had something to do with ...
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1answer
35 views

Infinite intersection between a arbitrary set of integers and a set of floor powers

Let $E$ be an infinite set of positive integers, proves that there is a $\alpha \in \mathbb{R}$ such that $\{\left \lfloor \alpha^k \right \rfloor ;k \in \mathbb{N} \}\cap E$ is infinite. I have two ...
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1answer
44 views

Equivalence of Forms of Baire Category Theorem

I am trying to show the equivalence of two forms of the Baire Category Theorem. These are the two statements: Let $(X,d)$ be a complete metric space. Let $U_n$ be a dense, open set for each $n \in ...
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For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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1answer
34 views

Convergence of a sequence holomorphic functions

Let $f_n\in\mathcal{O}(\Omega)$ be a sequence of holomorphic functions, s. th. $f_n\rightarrow f$ pointwise in $\Omega$. Show that exist open and dense set $\Omega'\subset \Omega$ such that $f_n$ is ...
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121 views

Showing the sum of two sets contains an interval - Baire's Theorem?

If $E$ and $F$ are measurable subsets of $\mathbb{R}$ and $m(E), m(F) >0,$ then $E+F$ contains an interval. The path to the standard solution to this is built on the notion that a measurable ...
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57 views

Open mapping lemma - are these versions equivalent?

Here is a version the Open Mapping Lemma given in class : Let $X$ be a Banach space and $Y$ be a normed space. Let $T : X\rightarrow Y$ be a bounded linear map. Assume there exist $M \geq 0$ and ...
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59 views

Strong version of Baire Category Theorem

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense. In such spaces is the intersection of fewer than $2^\omega$-many open ...
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109 views

Baire's property iff first category has dense complement.

Show that $(S, d)$ has Baire's property iff every set of first category has a dense complement. A set is of first category if it is a countable union of nowhere dense sets. First Category Baire's ...
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26 views

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 ...
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1answer
25 views

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 ...
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137 views

baire category and the union of dyadic balls of rational center

Suppose that $\{r_n\}_{n=0}^\infty$ is an enumeration of $\mathbb{Q}^N$ and $U = \bigcup_{n=0}^\infty B(r_n,2^{-n})$. We can use a trivial measure theory argument to prove that $U \neq \mathbb{R}^N$. ...
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27 views

Continuity Points of The Deriviative of a Differentiable Function

I recently read the section on Baire Spaces from Munkres and this question came to my mind which I cannot settle. Does there exist a differentiable function $f:\mathbf R\to\mathbf R$ such that the ...
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1answer
32 views

For $f :\mathbb R \to \mathbb R $, there exists an $(a,b)$, such that $f$ is bounded on a sequence with limit $x$, for all $x\in(a,b)$

I want to prove the following. Let $f : \mathbb R \mapsto \mathbb R $. Show that there exists an interval $(a,b) \in \mathbb R $ and $c >0 $: such that for any $x \in (a,b) $ there is a sequence ...
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179 views

Proof that every closed subset of $\mathbb R$ is finite or countable or continuum.

I want to prove that every closed subset of $\mathbb R$ is finite or countable or continuum. I know that for arbitrary subset we can not make similar statements - because of continuum hypothesis. ...
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how to show boundary of either open or closed set is nowhere dense.

how to show boundary of either open or closed set is nowhere dense. i think we need to use baire category thm? countable intersection of dense, closed set is once again a dense, closed (and ...
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33 views

If $\mathbb{R}=\bigcup_{n=1}^{\infty}E_n$, then the closure of some $E_n$ contains an interval

I'm working on some problems in Carothers' Real Analysis. I just started the section on the Baire Category Theorem. Thus far Carothers has given the Baire Category Theorem for $\mathbb{R}$. Prove ...
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222 views

Using Baire Category Theorem to prove $\mathbb{R}^2\not\cong\mathbb{R}^3$.

How can we prove $\mathbb R ^ 2$ is not homeomorphic to $\mathbb R ^3$ using Baire Category Theorem? Here is a standard proof of this fact using algebraic topology. Note that $\mathbb{R}^{3}-\{x\}$ ...
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Baire related problem

Let $f:\Bbb R\to \Bbb R$, $f$ in $C^{\infty}$. Suppose that for all $x \in \Bbb R$, there exists an integer $n$ (which depends on $x$) such that $f^{n}(x) = 0$ ($f^{n}$ is composing $f$ $n$ times) ...
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38 views

Non-decidable $\Pi^0_1$ (effectively closed) classes

Are there non-decidable $\Pi^0_1$ (effectively closed) classes? According to a draft of Effectively closed sets by Cenzer and Remmel, the class $$ P = \{ 0^n1^\omega \mid n \in B\} $$ is a ...
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1answer
51 views

Is the union of all $l^p$ spaces meagre in $l^\infty$?

Is the union of all $l^p$ spaces meagre in $l^\infty$? i.e. is $ \bigcup_{p=1}^\infty l^p$ meagre? I am revisiting this variety of math after a long break so help is appreciated. Please correct ...
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52 views

Is $c_0$ meagre in $l^\infty$?

I have been reading up on the meagre sets sans an instructor after a 10 year break from this sort of mathematics and I would like to test my understanding. I want to know if the following is true (and ...
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2answers
104 views

Showing a set is nowhere dense in $C([0,1])$

Let $E_n$ be the set of all $f \in C\big([0,1]\big)$ for which there exists $x_0 \in [0,1]$ (depending on $f$) such that \begin{align*} \lvert\, f(x)-f(x_0)\rvert \leq n\lvert x-x_0\rvert, ...
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69 views

A Property of Baire Spaces

Let $X$ be a topological space. I define $X$ to have Property A provided that every closed meager subset of $X$ is nowhere dense. It is easy to see that all Baire spaces have Property A. Is the ...
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1answer
246 views

In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?

I was playing with Baire's Theorem, and seemed to deduce the following: In a complete metric space $X$ that has no isolated points, any countable intersection of open dense sets is uncountable. ...
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1answer
51 views

A homeomorphism of a topological space with itself maps a set into one of the same category

Prove: If $h$ is a homeomorphism of $S$ onto $S$ and if $E\subset S$, then $E$ and $h(E)$ have the same category in $S$. Rudin, Functional Analysis, 2/e, p.43. (My own answer follows below.)
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Baire Category for monotonic sequence

Let X be a non-empty complete metric space and let $\{f_n:X\to \Bbb R\}^\infty_{n=1}$ be a sequence of continuous functions with the following property: for each $c\in X$, there exists an integer ...
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Is the complement of a set 1st category set $X$ of 2nd category?

If I have some metric space $M$ and find that $X \subset M$ is of 1st category (resp. 2nd category), is the complement of $X$, $X^c$ of second cateogry (resp. 1st category)? Thanks!
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How can the Baire Category Theorem be used to show a point in a complete metric space has some particular property?

My book gives a very tantalizing yet brief note that the Baire Category Theorem can be used to show that a point $x$ in a complete metric space $M$ has a particular property $P$. It states: "A ...
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1answer
36 views

A set that is a countable intersection of open and dense sets but not open.

We know that Baire Category Theorem implies that in a complete metric space, the countable intersection of open and dense sets is nonempty and actually dense itself. But it is clear that a countable ...
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1answer
83 views

Why is the subspace of polynomials with degree $\leq$ n nowhere dense in $\mathbb{R}[X]$?

There's a popular application of Baire's Category Theorem that shows that $\mathbb{R}[X]$ (the space of all polynomials with real coefficients) is not complete by showing it is a countable union of ...
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Baire's Theorem Formulation

I am still having problems with some of the proofs of Baire's theorem. In Introductory Real Analysis by Kolmogorov and Fomin The statement of the theorem states that: "A complete metric space R ...
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Is complement of a 1st category set in a 2nd category space, dense?

Let $X$ be a second category space and $Y$ be a first category subspace of $X$. Is $X \setminus Y$ dense in $X$?
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55 views

B meager, not empty, not open implies complement is dense

Let X be topological space. The subset B is meager, not open and non-empty. How can you prove that the complement is dense?
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52 views

Baire's Theorem proof regarding points revisited

My first question on this point was not answered. Here is the first part of Shilov's proof of Baire's theorem (not an exact lift from the book as I avoided mathematical symbols). I am trying to be ...
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1answer
50 views

Baire category theorem to show a set is dense.

Consider $A_j$ a sequence of subsets of $[0,1]$ s.t. for each $N\geq 1$, $\bigcup_{j=N}^\infty A_j$ is open and dense in $[0,1]$. If $S$ is the set of points $x \in [0,1]$ s.t. $x \in A_j$ for ...