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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 $X$...
Noah Schweber's user avatar
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 $\sum a_n$ ...
Boaz Tsaban's user avatar
  • 3,104
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 $F$ ...
Alexander Osipov's user avatar
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 ?
Alexander Osipov's user avatar
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 ...
Alexander Osipov's user avatar
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 ...
Boaz Tsaban's user avatar
  • 3,104
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. ...
Alexander Osipov's user avatar
4 votes
1 answer
353 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 $X$ ...
Boaz Tsaban's user avatar
  • 3,104
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\...
Alexander Osipov's user avatar
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\...
Alexander Osipov's user avatar
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 $...
Alexander Osipov's user avatar
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 ...
Alexander Osipov's user avatar
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}\...
Dans0804's user avatar