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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on a countably union of $F_\sigma$ sets

Let $F$ be a countably union of $F_\sigma$ sets, say $F_i$ for $i\in \mathbb N$. Is it true that if $F$ is not meager then there is an $F_i$ with non-empty interior? For me it's true but I can't find ...
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1answer
34 views

Every comeager set in a perfect Polish space contains an uncountable dense $G_\delta$ set

Let $X$ be a perfect Polish Space. Prove that every comeager contains an uncountable dense $G_\delta$ set. It's known that every perfect Polish Space has cardinality $2^{\aleph_0}$. It's easy to ...
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1answer
23 views

on comeager sets of a Baire Space

Is it true that in a Baire Space the intersection of two comeager sets is not empty? If yes, is the intersection comeager too?
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1answer
21 views

open dense subset of a Baire Space

Let $X$ be a Baire Space and let $Y$ be a comeager subset of $X$. Is true that $Y$ contains an open dense subset of $X$? Thank you
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1answer
17 views

is the complement of first category is always second category

is the complement of first category is always second category in general space( which is not complete). I think it is true only if the space isx
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0answers
12 views

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G.

If A is nowhere dense in M, and if G is a nonempty open set in M, prove that A is nowhere dense in G. I tried by contradiction but could not figure it out. I found that we can use following result ...
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1answer
46 views

uncountable co-meagre set in Polish Spaces

Let $X$ be an uncountable Polish Space and let $Y$ be a co-meagre subset of $X$. How can I prove that $Y$ is uncountable? Possibly proof without using borel sets. Thank you
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2answers
32 views

Zero-dimensional separable metric spaces

I have to prove that every separable metric space, which is zero-dimensional is isomorphic to a closed subset of the Baire space. Maybe I can use the Baire category theorem, but I don't know how.
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0answers
40 views

Proof of Invariance of Domain Using Baire Category and Local Homology

$\newcommand{\R}{\mathbf R}$ Claim. Let $m\neq n$ be positive integers. Can there exist a bijective continuous map $f:\R^m\to \R^n$? I think the answer is no and following is my argument. Purported ...
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3answers
95 views

If G is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in R. is it true in general metric space?

If $G$ is open and dense subset in $\mathbb R$ then show that $G\setminus\{x\}$ is also open and dense in $\mathbb R$. is it true in general metric space? I know as $G$ is open and singleton set ...
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0answers
15 views

Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense ...
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1answer
24 views

Empty interior, equivalent definitions from Munkres.

The Munkres book states the following definition: Recall that if $A$ is a subset of a space $X$ the interior of $A$ is defined as the union of all open sets of $X$ that are contained in $A$. ...
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1answer
31 views

Continuous image of a F-space

Let $X$ be a $F$-space, i.e. there exists complete metric $d$ on $X$ such that $d(x,y)=d(x+z,y+z), \ x,y,z\in X$. Suppose that we have a normable vector space $Y$, which is of first Baire category. ...
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0answers
85 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in ...
3
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2answers
123 views

Baire Category Theorem & The Real Numbers

I am taking a Real Analysis unit at University and the topic of the Baire Category Theorem is prevalent in all of the course, however I'm actually, embarrassing stuck right at the start of the ...
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2answers
335 views

Contradiction to Baire's Theorem?

My book states a version of Baire's Theorem as If $\{G_1,G_2,G_3,\ldots\}$ is a countable collection of dense, open sets, then the intersection $\bigcap_{n=1}^\infty G_n$ is not empty. I am ...
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1answer
29 views

The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
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1answer
38 views

Is the product of second category spaces second category?

The following is an exercise of Gemignani's Elementary topology. (I'm studying for an exam.) Suppose $Y$ and $Z$ are subspaces of some space $X,\tau$ and $Y$ and $Z$ are both of the second category in ...
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0answers
56 views

prove that there is a dense subset of X on which $f$ is continuous.

Let ${(f_n)}$ be sequence of continuous function on a complete metric space X which converges point-wise to a function $f$ then prove that there is a dense subset of X on which $f$ is continuous. I ...
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1answer
35 views

Problem seeing how Baire's theorem applies in a proof of open mapping theorem

I cant see which version and how they use Baires theorem to get that atleast on $MB_{n}$ is dense in some open set. Any version of Baires theorem needs open or closed sets. I can get neither on the ...
2
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1answer
34 views

$G_\delta$ set of nowhere differentiable functions?

I'm trying to show that in $C([0,1])$ with the supremum metric, there exists a dense $G_\delta$ set of nowhere differentiable functions. Honestly, I don't know how to approach this. Any help would be ...
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1answer
83 views

Nowhere dense subset of $L^1$

Why is $B_n = \{f \in L^1 : \int |f|^2 < n \}$, $n \in \mathbb{N}$ a nowhere dense subset of $L^1$? Please provide a proof without assuming that $L^2 \subsetneq L^1$. Clarification: $L^p$ here ...
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1answer
21 views

Closed set of 1st category

If $F$ is a closed subset of a complete metric space, is it possible for $F$ to be of the first category? This seems to lead to a contradiction. Since $F$ is a closed subset of a complete metric ...
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1answer
28 views

Show that C^α([0, 1]) is of first category in C([0, 1]).

Recall that for any $0 < α < 1,$ the space $C^\alpha ([0, 1])$ is the set of continuous functions on $[0, 1]$ with norm of f = sup |f| + $ sup \ x\ne y$ |f(x) − f(y)|/|x − y|^α< ∞, equipped ...
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1answer
97 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Question. Let $\mathbb A=\mathbb R\!\smallsetminus\!\mathbb Q$ be the irrational numbers. Is there a continuous function $\,f:\mathbb A\to\mathbb Q$, which is nowhere locally constant? – i.e., for ...
2
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1answer
37 views

Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense.

Let $f: \mathbb R \rightarrow \mathbb R$ be a continuous function that is nonconstant on any interval. Prove that if B is closed and nowhere dense, $f^{-1}(B)$ is also closed and nowhere dense. My ...
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2answers
38 views

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 ...
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1answer
43 views

Baire category of subset $\mathbb{R} \setminus B$ with lebesgue zero measure $B$

Let $\mathbb{Q}=\{ q_i\}^{\infty}_{i=1}$. Let $I_{ij} = \left(q_i - \delta_{ij}, q_i + \delta_{ij}\right)$, $\delta_{ij}=2^{-i-j-1}$ for all $i,j \geq 1$ be an open interval. For every $j\geq1$, ...
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1answer
52 views

Baire's Theorem and Irrationals

I am asked to show that the irrational numbers are not a countable union of closed subsets of $\mathbb{R}$ given that if a complete metric space is the countable union of of closed subsets then at ...
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0answers
59 views

Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...
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1answer
55 views

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 ...
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2answers
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Proving a result in Baire categories

If $A\subset X$ is a $G_\delta$ set and dense in $X$, then what I want to show is that $X\setminus A$ is of the first Baire category. Where the meaning of first category is that I can write ...
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1answer
45 views

On the zero set of a $C^2$ function on $[0,1]^2$ [closed]

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 ...
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1answer
41 views

continuous an open image of a meager set is meager

I want to know if the following is true. Let $X$ and $Y$ be topological spaces and $f\colon X\to Y$ a continuous open surjection. Suppose that $X$ is meager, then $Y$ is meager. Recall that a meager ...
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1answer
51 views

Question related to Baire category theorem

I met this question from a faculty booklet which I had trouble on involving metric spaces, stating: Let $ (X,d) $ be a complete metric space with the sequence of closed sets, $ \{F_n\}_{n \in N} $ ...
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1answer
54 views

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 ...
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1answer
14 views

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 ...
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1answer
23 views

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. ...
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1answer
29 views

How does this follow from the Baire category theorem?

The book says that statement 2 is a direct consequence of statement 1. I don't see how they prove statement 2 directly from statement 1, can you please help me? Statement 1: A complete metric ...
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1answer
44 views

Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials

Problem statement: Show that $R^2$ cannot be written as a countable union of zero sets of non-trivial polynomials. Note that the zero set of a polynomial $p(x,y)$ is $\{(x,y) : p(x,y) = 0$}. My ...
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1answer
32 views

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 $E$ ...
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1answer
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Second Baire class and Borel measurable

I am new to Baire class theory, but need it for one part of a project I am working on. I have seen it referenced that functions of second Baire class are Borel measurable. For example here in this ...
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0answers
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Lebesgue characterization of Baire class 1 functions

Lebesgue's characterization of Baire class one functions on $\mathbb R$ is the following: $f:\mathbb R \rightarrow \mathbb R$ is Baire class one iff for all $r\in \mathbb R$ $\{ x: f > r \}$ and ...
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1answer
85 views

Application of Baire Category theorem

Suppose that $f$ is infinitely differentiable on $[a,b]$ and suppose that for any $a ≤ x ≤ b$ the Taylor series of $f$ has positive radius of convergence at $x$. Use the Baire Category Theorem to ...
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1answer
42 views

Is $C[0,1]$ equipped with $\lVert \cdot \rVert_1$ a countable union of nowhere dense sets?

Let's consider the space of all continuous function $C[0,1]$ on the intervall $[0,1]$. But instead of using the usual supremum norm we use the $L^1$-Norm $: \lVert f \rVert_1=\int_0^1 \lvert f(x) ...
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1answer
94 views

Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
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1answer
75 views

Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function such that for every $x \in \mathbb{R}$ there exists $n$ such that $f^{(n)}(x) = 0$, then f is a polynomial. I'm kind of lost on this ...
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1answer
95 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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2answers
52 views

Undestanding an assumption in Baire's theorem proof.

I was reviewing the proof of Baire's theorem I saw in class a few days ago, and there's an assumption that I didn't managed to see where it was used. I'll put here the proof given. Theorem (Baire) ...
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1answer
31 views

Is the closure of a meager set meager?

How to show that the closure of a meager set is meager? I tried like this: Suppose that it is not meager then $cl(A)$, where $A$ is a meager set in a metric space $(X,d)$ contains an interior point ...