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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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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1answer
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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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2answers
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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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1answer
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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
27 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
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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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2answers
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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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1answer
30 views

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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1answer
37 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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1answer
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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
45 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 ...
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0answers
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how do i prove that the the set of irrationals cannot be a countable union of closed subsets? [duplicate]

Let $\mathbb{R}$ be equipped with the standard topology. Let $E$ be the set of irrational numbers. How do i prove that $E$ is not a countable union of closed subsets, using Baire Category Theorem?
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0answers
29 views

Showing an operator kernel is nowhere dense…

We are doing a problem that requires us to, for fixed NONZERO element $h \in L^2[0,1]$ (as in, the function $h$ is not zero on a set of positive measure in $[0,1]$) and fixed $f \in L^2[0,1]$ with ...
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1answer
96 views

Nowhere dense set…

Let $A_n$ be a subset of continuous functions on $[0,1]$ given by: $A_n$ = {$f∈C[0,1]$:there exists $x∈[0,1]$ such that $|f(x)−f(y)|≤n|x−y|$ for all $y∈[0,1]$}. Show $A_n$ is nowhere dense, and use ...
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0answers
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Uniform boundedness… [duplicate]

Let $\{f_n\}$ be sequences of real valued continuous functions on $\Bbb R$ that is convergent pointwise on $\Bbb R$. Show that there exists $M>0$ and interval $I\subset \Bbb R$ such that ...
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1answer
91 views

A Baire category question

Let ${f_n}$ be a sequence of real valued continuous functions converging pointwise on $\Bbb R$. Show that there exists a number $M>0$ and an interval $I \subset \Bbb R$ such that $\sup\{ |f_n(x)|:x ...
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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 ...
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If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
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1answer
46 views

Linear functional in Banach space

Let $X$ be a Banach space, $(f_n)\in X^{*}$ a sequence with $f_n\neq 0 $ $ \forall n\in \Bbb N$. Show that there is a $x\in X$ such that $f_n(x)\neq 0 $ $\forall n$. Need some help. Thank you!
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1answer
60 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
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0answers
28 views

Analogy for Baire categories?

I'm looking for an analogy to grasp the intuitive notion of size that Baire categories on $\mathbb{R}$ provides. For instance, the cardinality of a subset of $\mathbb{R}$ provides a notion of size in ...
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1answer
44 views

Equivalence of Baire Space definitions

I am hoping someone could help me show that the following statements, which define a Baire Space, are equivalent. Defn1: Any topological space X such that the intersection of any countable ...
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0answers
48 views

Kuratowski-Ulam Theorem

I am looking for a proof of the Kuratowski-Ulam Theorem: Let $X,Y$ be Baire spaces, $Y$ be second countable and let $R \subseteq X \times Y$ have the Baire property. Then $\{ x \in X \ | \ R_x ...
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1answer
109 views

Decomposing real line as a union of a nullset and a set of first category

$\Bbb R$ can be written of the form $A\cup B$ such that $A$ is of measure zero and $B$ is of the first category! can anybody prove this?? I guess $A$ must be an $G_{\delta}$ set which is dense in ...
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1answer
33 views

Application of Baire's theorem

Let $f: (a,b) \rightarrow \Bbb R$ be a differentiable function in $(a,b)$. Calculate the pointwise limit of: $$f_n(x)=n(f(x+1/n)-f(x)), x\in(a, b-1/n). $$ Let $E_n$ be a countable ...
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0answers
17 views

No Baire measurable reduction from $\mathbb{E_0}$ to the diagonal of a second countable Hausdorff Space

On $2^{\mathbb N}$ we define the equivalence relation $x \mathbb{E_0} y \text{ iff } \exists m \in \mathbb N \forall n \ge m: x(n) = y(n)$ and notice that every $\mathbb E_0$-invariant set with the ...
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2answers
42 views

Baire space but not locally compact

I need an example that is a Baire space but not locally compact. I think, $\mathbb{R}^ \mathbb{R}$ is such an example. $\mathbb{R}^ \mathbb{R}$ is not locally compact. But I could not proof that it ...
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3answers
43 views

Meagre Sets: Algebra

Let meagre subsets be defined as: $A\text{ meagre}\iff A=\bigcup_{\lvert K\rvert\leq\aleph_0} A_k\text{ with }\overline{A_k}°=\varnothing$ Then it satisfies: $B\subseteq A\text{ meagre}\Rightarrow ...
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1answer
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Baire: Equivalent Statements II

Moreover, why does it follow for Baire Spaces and why is it strictly weaker that: $X=\bigcup_{k=1}^\infty A_k\quad\Rightarrow\quad\exists k_0\in\mathbb{N},x_0\in X: A_{k_0}\in\mathcal{N}_{x_0}$ ...
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2answers
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Baire: Equivalent Statements

How to proof that these statements are equivalent: Every intersection of countably many dense open sets is dense. The interior of every union of countably many closed nowhere dense sets is empty. ...
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1answer
49 views

Prove that a set is dense and of the first category in $L^2(T)$

Define the Fourier coefficients $\hat{f}(n)$ of a function $f\in L^2(T)$, $T$ is a unit circle, by: $\hat{f}(n)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}f(e^{i\theta})e^{-in\theta}d\theta$ Let ...
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2answers
219 views

Irrational number and Baire space

How to show that the set of irrational numbers is a Baire space ?
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1answer
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Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
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1answer
59 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
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1answer
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Baire Category Theorem proof in Gamelin Greene - how do they shrink the closure of open ball

I am confused by a step in the Gamelin and Greene proof of the Baire Category Theorem. Here is the start of the proof. Theorem: Let $\{U_n\}_{n=1}^{\infty}$ be a sequence of dense open subsets of a ...
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1answer
70 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
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1answer
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Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
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1answer
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Baire category related question

Let $A_n$ be a sequence of closed sets of $\mathbb{R}$ such that $[a,b]\subseteq \cup A_n$, for some $a<b $. Show that at least one of the set $A_n$'s contains an interval. Thanks in advance,
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1answer
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Oxtoby - Thm 6.2 - There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category

I am reading the book Measure and Category of Oxtoby. In Theorem 6.2 it is stated that There exists a strategy in which (A) wins iff $I_1 \cap B$ of first category As in the picture below. My ...
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2answers
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Who are the measurable sets in $\mathbb{R}$

Is there any Characterization for all measurable sets in $\mathbb{R}$? Can I say that a set is measurable if an only if it has the property of Baire? (differs from an open set by a first category ...
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1answer
88 views

Why any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset homeomorphic to $[0,1]$

as a continuation to my question here: Is cantor set homeomorphic to the unit interval? I can't see how can it is be true that any uncountable $G_\delta$ set of $\mathbb{R}$, has a subset ...
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1answer
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Baire's theorem proof

> A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets. I don't understand the following point the proof: we assume that $X$ is the countable union of ...
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3answers
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Is a Baire Space necessarily complete?

A complete metric space a Baire space. But is a Baire space necessarily complete?
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1answer
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Baire Category Theorem: What should we really prove there?

I am reading about the Baire Category Theorem in Jech's book on set theory. 4.8: Baire Category Theorem: Let $D_0,D_1,\dots,D_n,\dots$, $n \in \mathbb{N}$, be open dense subset of $\mathbb{R}$. Then ...
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1answer
134 views

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 ...
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1answer
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definition of Baire first category set

Suppose that $A$ is of the first category. Is it possible to write $A$ as a countable union of sets that are not necessarily nowhere dense? I mean after all a first category set is defined such that ...
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1answer
113 views

Prove it doesn't exist any function f:R→R that is continuous only at the rational points.

Prove it doesn't exist any function $f:\mathbb R \to \mathbb R$ that is continuous only at the rational points. Suggestion: For every $n \in \mathbb N$, consider the set $U_n=\{x \in \mathbb R : ...
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1answer
160 views

Examples of rare, meager and nonmeager sets in $\mathbb{R}$

Kreyszig Functional Analysis book presents the following definition. I'm trying to get some examples. (a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so ...
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Baire Category Theorem with $G_{\delta}$

If we have a subset $Y$ of a complete metric space $X$ such that $Y$ is a $G_{\delta}$, then how would I go to show that $Y$ satisfies the Baire category theorem? I am trying to get there by showing ...
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1answer
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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 = ...