# 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.

81 views

### 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 ...
102 views

### 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 ...
104 views

### 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 ...
66 views

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

Let ${\left(f_n\right)}$ 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 ...
53 views

31 views

### 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 ...
8k views

### Is $[0,1]$ a countable disjoint union of closed sets?

Can you express $[0,1]$ as a countable disjoint union of closed sets, other than the trivial way of doing this?
19 views

20 views

### 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 ...
234 views

25 views

### How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $G\subseteq SO(3)$ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
20 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 ...
45 views

### Why if $p (x_1, \dots, x_n)$ is polynomial on $\mathbb{R}^n$, then $p (x) \neq 0$ is satisfied by open dense set?

I have problems in seeing what exactly is the all point of first category and second category sets. Finally, I've found a reference (Bredon's "Topology and Geometry") that introduces the topic in a ...
78 views

### how to show boundary of either open or closed set is nowhere dense.

how to show boundary of either open or closed set is nowhere dense. i think we need to use baire category thm? countable intersection of dense, closed set is once again a dense, closed (and nonempty)...
6k views

### Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.

Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable. Can anyone help how can I solve the above problem?
17 views

### Proving Banach space is finite in Baire space [duplicate]

Let $B$ be a Banach space where the dimension of the underlying vector space is countable. using the Baire category Theorem, prove that the dimension of the underlying vector space is, in fact, finite....
### $X,Y$ metric spaces , $X$ complete , $Z$ is Hausdorff , $f,g:X \times Y \to Z$ continuous in each variable and coincide on a dense subset , is $f=g$?
Let $X,Y$ be metric spaces , $X$ be complete and $Z$ be a Hausdorff space ; let $f,g:X \times Y \to Z$ be functions such that each of $f$ and $g$ is continuous in $x \in X$ for each fixed $y \in Y$ ...