All Questions
8 questions
11
votes
2
answers
483
views
The "strong" measure number
Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, ...
- 20.5k
9
votes
0
answers
252
views
Another determinacy-related cardinal characteristic
This question is a kind of "dual" to an earlier one of mine.
Although I don't know a reference for this, it's easy to show the following result:
Suppose $G$ is a game in which neither ...
- 20.5k
9
votes
0
answers
243
views
Is this cardinal characteristic trivial? (Number of strategies needed to guarantee at least one win)
(Previously asked at MSE.)
Let the determinacy number, $\mathfrak{g}$ (for "game"), be the smallest cardinal such that for every (two-player, perfect-information, length-$\omega$) game on $\...
- 20.5k
8
votes
1
answer
351
views
"Compactness length" of Baire space
Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...
- 20.5k
6
votes
1
answer
258
views
Do escaping sets "uniformly" cover dominating sets under determinacy?
For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$...
- 20.5k
4
votes
1
answer
221
views
Comparing bornologies for cardinal characteristics via Borel maps
This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
- 20.5k
4
votes
1
answer
151
views
Comparing bornologies for domination/escaping
Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions from $\mathbb{N}$ to $\mathbb{N}$:
$\mathbb{D}=\{A: \exists f\in\mathcal{N}\forall g\in A\exists m\...
- 20.5k
4
votes
0
answers
293
views
How does "spreading-with-determinacy" compare with Cichon's diagram?
For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say that $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that $F[A]\in\mathbb{B}$...
- 20.5k