All Questions
16 questions
14
votes
1
answer
272
views
Is there a countably infinite closed interval in the lattice of topologies?
Is there an interval of the form $[\sigma,\tau]$ in the lattice of topologies on some set $X$ such that $|[\sigma,\tau]| = \aleph_0$?
In other words, do there exist two topologies $\sigma$ and $\tau$ ...
- 18.5k
14
votes
3
answers
1k
views
Order homomorphism functions on $\omega_1$
I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...
- 1,375
11
votes
0
answers
172
views
Can the nowhere dense sets be more complicated than the meager sets?
Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
- 18.5k
10
votes
0
answers
498
views
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals:
$\mathfrak p$ is the ...
- 1,101
8
votes
0
answers
241
views
Topological applications of $\mathfrak{p}=\mathfrak{t}$
I'm working on the Malliaris-Shelah's recent result of $\mathfrak{p}=\mathfrak{t}$, but I'm more interested in what possible topological applications can derivate from this equality.
Searching in ...
- 181
7
votes
1
answer
432
views
Existence of a specific mad family
Preliminaries:
Let $[\omega]^{\omega}$ be the set of all infinite subsets of $\omega$, the first countable ordinal (the set of the natural numbers).
We say that $\mathcal A\subset [\omega]^{\omega}$...
user46747
7
votes
0
answers
369
views
Baire category of tall ideals
Problem. Is it consistent with ZFC that $\mathfrak t=\omega_1$ and each $\omega_1$-generated tall $P$-ideal is of the second Baire category?
(Asked 01.10.2016 by David Chodounsky at page 20 of Volume ...
- 4,535
5
votes
1
answer
206
views
"König's theorem" for $T_2$-spaces?
For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
- 48.9k
5
votes
1
answer
155
views
Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?
Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...
- 41.9k
5
votes
0
answers
170
views
Can maximal filters of nowhere meager subsets of Cantor space be countably complete?
Let $X$ denote Cantor space. A subset $A\subseteq X$ is nowhere meager if for every non-empty open $U\subseteq X$, we have $A\cap U$ non-meager. We call $\mathcal{F}\subseteq \mathcal{P}(X)$ a maximal ...
- 369
5
votes
1
answer
241
views
On filters possessing a countable network
Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$
A family $\mathcal N$ of subsets of $\omega$ is called a network ...
- 41.9k
4
votes
0
answers
127
views
An uncountable Baire γ-space without an isolated point exists?
An open cover $U$ of a space $X$ is:
• an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$.
• a $\gamma$-cover if it is infinite and each $x\...
- 1,101
3
votes
0
answers
209
views
Nowhere Baire spaces
Studying the article "Barely Baire spaces" of W. Fleissner and K. Kunen, using stationary sets, they show an example of a Baire space whose square is nowhere Baire (we call a space $X$ nowhere Baire ...
- 561
2
votes
0
answers
200
views
A question about infinite product of Baire and meager spaces
Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space.
Does anyone have any suggestions to demonstrate Proposition 1?
I was ...
- 561
1
vote
1
answer
278
views
Borel hierarchy and tail sets
Let $A$ be a finite set, and let $A^\infty$ be the set of all sequences $(a_n)_{n=1}^\infty$ of elements of $A$.
A set $B \subseteq A^\infty$ is a tail set if for every two sequences $\vec a, \vec b \...
- 745
1
vote
0
answers
102
views
Functions preserving almost disjoint of partitions
A collection $\mathcal{A}\subseteq \omega^\omega$
is almost disjoint iff
$\bigcap_{X\in \mathcal{A}}X^{-1}(j)$ is finite for all $j\in\omega$.
A function $\Gamma:2^\omega\rightarrow 2^\omega$ is
...
- 909