# Tagged Questions

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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### Non empty interior in the image implies open map

I am looking at the proof showing that $L^2(0,1)$ is meager in $L^1(0,1)$. Define $B_n = \{f\in L^2 : \|f\|_2 \leq n\}$. With the continuous identity map $T:L^2 \rightarrow L^1$, if one of $T(B_n)$ ...
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### Proving the Baire Category Theorem from scratch, Stuck!

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other ...
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### Filter, which does not have the Baire property

I do not understand the following proof of the following theorem. It is part of Theorem 4.1.2 (Talagrand theorem) in the book Tomek Bartoszyński and Haim Judah: Set theory. On the structure of the ...
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### Meager subset of $2^\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
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### Paths in a connected subset of $\mathbb R^n$

Prove that every connected subset $X$ of $\mathbb R^n$ with more than one point has the continuum cardinality. In order to use Baire Lemma, I would like to demonstrate that there is a compact (or ...
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### Why for every rational $p$ the level sets $\{ f \leq p \}$ and $\{ f \geq p \}$ are $\prod_{\xi+1}^0$ sets?

$f$ is a real-valued baire class $\xi$ function if $\{ f < c \}$ and $\{ f >c \}$ are in $\sum_{\xi +1}^0$ for every $c \in \mathbb{R}$. In the proof of Proposition $5.2$ page $24$, we have ...
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### Shift of first category set in a compact metric space.

Question. If $X$ is a homogeneous compact metric space, and $F=\bigcup _{n\in\omega}F_n$ is a countable union of closed nowhere dense subsets of $X$, then is there a homeomorphism $\varphi:X\to X$ ...
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### Proof that a space is a Baire space.

Let $(Z,T_2)$ be a subspace of $R^2$ given by $Z = \{\langle x,y\rangle : x,y \in \Bbb R, y > 0\} \cup \{\langle x,0\rangle : x \in \Bbb Q\}$. Show that this is a Baire space. It is suggesting to ...
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### Borel set as union of $G_\delta$ and countable set

What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?
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### What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
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### Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, ...
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### $G_\delta$ 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 ...
### 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 ...