# 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.

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### Baire space and increasing union of closed subspaces

Let $X$ be a Baire space. Suppose there is an increasing sequence $C_1\subset C_2\subset \cdots$ of closed subspaces of $X$, whose set-theoretical union is $X$. Since $X$ is Baire, we know that some ...
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### The set is meager if it has a cover of clopen meager sets

Let $X$ be a topological space such that there exists a collection of meager clopen sets $(C_i)_{i \in I}$ such that $X = \bigcup_{i \in I} C_i$. I want to prove that $X$ is then meager itself. As ...
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### Does a $G_\delta$ set of second category always contain an $F_\sigma$ set of second category?

We know that any $F_\sigma$ subset of $\mathbb{R}$ of second category always contains a $G_\delta$ set of second category, because an $F_\sigma$ set has the property of Baire and therefore can be ...
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1 vote
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### Countable sum of Baire of class 2 functions on $\Bbb R$

For every $k\in\Bbb Z$, let $f_k$ be a Baire 2 class function on $\Bbb R.$ Assume $\sum_{k\in\Bbb Z} f_k$ is convergent. Define $f:=\sum_{k\in\Bbb Z} f_k$ so $f$ is a function. Moreover, $f$ is a ...
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### Subset with only digits $0$ and $1$ in their decimal expansion

Consider $[0, 1]$ with its Euclidean metric and the set $A ⊆ [0,1]$ of real numbers $x$ that contain only digits $0$ and $1$ in their decimal expansion. It is clear to me why this set is not dense : ...
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### More characterizations of Baire space

We know there are a bunch of equivalent definitions of Baire space. For example: Given a metric space $M$. 1.The complement of 1st category set in $M$ is dense in $M$. For more equivalent definitions, ...
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### Can we classify all topological space $(X, \tau)$ where every second category sets are Residual sets?

$(X, \tau)$ be a topological space. $A\subset X$ is Residual if $X\setminus A$ is of first category. In a Baire space, a Residual set is of second category. $A\subset X$ Residual, then $X\setminus A$ ...
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### If a Banach space has a countably infinite basis, then it has finite dimension

I'm solving Ex 3.8.2. in Brezis' book of Functional Analysis. Could you have a check on my attempt? Let $(E, | \cdot |)$ be a Banach space. Let $(e_n)$ be a countable basis of $E$, i.e., each $x\in E$...
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### Show that a set of continuity points has an open dense subset

Let $f_n:\mathbb{R}\to\mathbb{R}$ be continuous maps, $\forall n\in\mathbb{N}$, and $f_n(x)\to f(x),\forall x\in\mathbb{R}$. Let $\varepsilon>0$. Show that there there is an $N\in\mathbb{N}$ and a ...
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### Why was the concept of first/second categories in metric spaces introduced?

Let $(X, d)$ be a metric space. $X$ is of first category if it can be expressed as a countable union of nowhere dense subsets. Otherwise, $X$ is called a metric space of second category. A first ...
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### Uncountable subset $X$ of $\mathbb{R^{n}}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$.

$\textbf{Question}$: Show that there exist an uncountable subset $X$ of $\mathbb{R}^{n}$ with property that every subset of $X$ with $n$ elements is a basis of $\mathbb{R}^{n}$. $\textbf{My Attempt}$: ...
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### Show if a family of maps if bounded pointwise, then it is bounded uniformly on a dense subset of an open ball

Let $M',M$ be metric spaces with $M$ a Baire space. If any family $(f_i)_i$ of maps $M\to M'$ is bounded pointwise on $M$, then it is also bounded uniformly on a dense subset of an open ball in $M$. ...
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### Question about dense subset of an Banach space

Let $E$ be an Banach space and $A \subset E$ an dense subset. It is possible to find a function $f:E \to \mathbb{R}$ such that, for each $x \in A$, $\lim_{t \to x} |f(t)|=\infty$? I don't know how to ...
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### Simple question about Open Mapping Theorem's proof in Functional Analysis

Regarding the OMT's proof, I have a really simple question, but I can't found an acceptable answer for it, every source simply ommits it (for this motive I think it's really simple). So, let's state ...
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### Baire space and Topological property

Show that the property of being a Baire space is a topological property . Baire space : In mathematics, a topological space X is said to be a Baire space, if for any given countable collection {An}...
1 vote
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### Complement of meagre set contains a dense $G_\delta$ set

While reading a Functional Analysis book, the autor stated, in a somewhat crude manner, that, if $X$ is a Baire space and $H \subset X$ is meagre, then $X \setminus H$ contains a dense $G_\delta$ set. ...
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### What is the justification for these two steps in proof of Baire's theorem?

Below is the statement and proof of Baire theorem from the textbook I'm using. I have some difficulties understanding it and I would appreciate if someone can explain it in more details. In particular ...
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### Is an NVS complete iff it is non meagre?

Let $V$ be an NVS. We know by Baires theorem that complete $\implies$ nonmeagre. What about the converse? Can we have a non complete space that is nonmeagre or not? For metric spaces the answer is yes ...
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### How to construct a real valued function which is continuous only on a given $G_\delta$ subset of $\mathbb{R}$?

I know that the set of continuity of a function $f:\mathbb{R} \to \mathbb{R}$ forms a $G_{\delta}$ subset of $\mathbb{R}$. My question is, given any $G_{\delta}$ subset of $\mathbb{R}$ , how can I ...
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### $A$ has property of Baire if and only if $A=B \sqcup Q$ where $B$ is a $G_{\delta}$ set and $Q$ is of first category

Over $\Bbb{R}$, A set $A$ with a property of Baire is defined in our notes as the symmetric difference $A=G\triangle Q$ where $G$ is open and $Q$ is of first category. I am asked to show that a set $A$...
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### The image of bounded linear operator is the whole space or first Baire category [duplicate]

Let $V$ be a complete normed space and $W$ is normed space. Let $A \subset V$ be closed subspace. If $T:V \rightarrow W$ is bounded linear operator, then we need to prove that $T(A)$ is either first ...
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### Topological transitivity definition

Let $X$ be a Baire space, $f$ a map $X \to X$, $x$ a point of $X$, $\omega (x,f)$ its limit set, $Trans(f)$ the set of transitive points. In many sources (e.g. here) the following definitions of "...
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