# 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.

66 views

### A function continuous on rational points and discontinuous on irrational points [duplicate]

How to find function $f : \Bbb R \to \Bbb R$ such that $f$ is continuous on the rational numbers and discontinuous at irrational numbers? I was told to use the Baire Theorem to show that the set of ...
110 views

### 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 ...
229 views

46 views

### Question about the Baire space, $\sigma$-algebra and $\sigma$-ideal.

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$. Assume $X$ is second countable Baire space....
73 views

### Question about of Baire property and Baire space

In reading Kechris book. Please, I would like help with this proposition. For convencion we put for $A \subseteq X$, $$\sim A=X\setminus A$$ If $A$ is comeager in $U$, we say that $U$ forces $A$, ...
81 views

### Q: Nowhere dense sets.

Given $X$ a metric space, $A\subset X$ a nowhere dense set. Show that every open ball $B$ contains another open ball $B_1 \subset B$ such that $B_1 \cap A = \emptyset$. EDIT: I modify my proof with ...
46 views

### intersection about the second category

$G$ is a locally compact Hausdorff topological group, $A$ and $B$ are two Borel subsets of $G$, and $A$ and $B$ are both of the second category in $G$, then there exist an element $x\in G$, such that ...
97 views

### Requirements for the principle of uniform boundedness

The version of the principle of uniform boundedness as we stated it in the lecture seems wrong to me in multiple points. Here is how I would state and proof the principle in the terms we used in the ...
68 views

### Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces

Show that $\mathbb{R}^n$ cannot be written as a countable union of proper subspaces Ok so I know I have to use Baire's Cathegory Theorem here. And I've done the following, lets suppose on the ...
48 views

### Topological group, which is second category in itself, is a Baire space.

A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. $G$ is a topological group, if $G$ is of the second ...
45 views

74 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 ...