All Questions
Tagged with descriptive-set-theory selection-principles
13 questions
9
votes
1
answer
428
views
The cardinality of projections of subsets of the Hilbert cube by inner products
I have three related questions.
Question 1: Is there a subset $X$ of the Hilbert cube $[0,1]^{\Bbb N}$ of cardinality continuum, such that for each sequence $a\in [0,1]^{\Bbb N}$ with $\sum a_n$ ...
- 3,104
6
votes
1
answer
149
views
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective?
Is there a Bernstein subset $X$ of $\mathbb{R}$ such that no continuous map $f : X → [0,1]$ is surjective ?
- 1,101
3
votes
1
answer
177
views
Is there a metric separable space with the following properties...?
Let $\omega_1<\mathfrak{q}_0$ where $\mathfrak{q}_0:=\min\{|Y|:Y\subseteq \mathbb{R}$, $Y$ is not a $Q$-space$\}$.
Is there a metric separable space $X$ with the following properties:
$|X|\geq\...
- 1,101
5
votes
1
answer
137
views
Is it consistent that the additivity of Lebesgue null sets is greater than $\frak h$?
This question concerns combinatorial cardinals of the continuum.
Some of these are listed in the following diagram, from Blass's survey on the topic.
There are some additional cardinals, related to ...
- 3,104
3
votes
0
answers
141
views
Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?
Which cardinal $\kappa\geq \omega_1$ is critical for the following property:
Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
- 1,101
2
votes
0
answers
171
views
Is there a Lusin space $X$ such that ...?
Is there a Lusin space (in the sense Kunen) $X$ such that
$X$ is Tychonoff;
$X$ is a $\gamma$-space ?
Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin.
In mathematics, a ...
- 1,101
4
votes
1
answer
352
views
Almost compact sets
Update:
Q1 is answered in the comments.
I think that the usual arguments show that every relatively almost compact set in a space is closed in the space.
Original question:
A set $K$ in a space $X$ ...
- 3,104
5
votes
1
answer
370
views
Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Recall that
$\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$.
The cardinal $\mathfrak{q}_0$ defined as the smallest ...
- 1,101
4
votes
1
answer
94
views
Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?
A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$.
A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$.
Definition. ...
- 1,101
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
1
vote
1
answer
148
views
Scales and concentration
Let $S$ be a dominating subset of $[\mathbb{N}]^{\infty}$. Then $S=\{s_{\alpha} : \alpha<\mathfrak{d}\}$ is called a $\mathfrak{d}$-scale if for every $\alpha<\beta<\mathfrak{d}$, $s_{\beta}\...
- 21
7
votes
1
answer
209
views
Are σ-sets preserved by Borel isomorphisms?
Recall that a $\sigma$-space is a topological space such that every $F_{\sigma}$-set is $G_{\delta}$-set.
$X$ - $\sigma$-set, if $X$ is a $\sigma$-space and it is subset of real line $R$.
Let $F$ ...
- 1,101
18
votes
1
answer
772
views
Two strengthenings of "strong measure zero"
A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering $X$...
- 21.2k