All Questions
34 questions
32
votes
1
answer
2k
views
Bidi: A new cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...
- 3,104
15
votes
2
answers
965
views
$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?
An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter.
Let $\mathfrak{ufo}$ be the minimal cardinality of
an ultrafilter ...
- 3,104
9
votes
2
answers
281
views
Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?
Is there a condensation (continuous bijective mapping) from $D^{\aleph_0}$ onto a metrizable compact space ?
$D$ - discrete space of cardinality $\aleph_1$.
CH implies it is a positive answer. In ...
- 1,101
9
votes
1
answer
264
views
Rothberger property for finite covers
Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...
- 41.8k
8
votes
1
answer
451
views
Strong measure zero sets and selection principles
A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
- 213
7
votes
1
answer
389
views
References for higher descriptive set theory surveys
A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
- 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 $F$ ...
- 1,101
6
votes
3
answers
655
views
When does the generalized Cantor space embed in a $\kappa$-compact space
The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...
- 3,104
6
votes
1
answer
772
views
A ridiculous combinatorial cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the ...
- 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
6
votes
1
answer
260
views
A ZFC example of a Menger space which is not Scheepers
$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
- 505
6
votes
1
answer
298
views
What is the height (or depth) of $[\mathbb{N}]^\infty$?
(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...
- 3,104
5
votes
1
answer
600
views
When is the generalized Cantor space $\kappa$-compact?
My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...
- 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
5
votes
1
answer
419
views
When is there an unbounded tower in $[\mathbb{N}]^\infty$?
(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...
- 3,104
5
votes
0
answers
320
views
Unbounded towers and combinatorial cardinal characteristics of the continuum
Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...
- 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. ...
- 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
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
3
votes
1
answer
294
views
Implications between different covering properties of spaces
Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set,...
- 48.9k
3
votes
1
answer
165
views
Menger and Scheepers subsets of $\mathbb R$
$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
- 505
3
votes
1
answer
135
views
Characterization of the Scheepers property by Scheepers game
$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
- 505
3
votes
1
answer
263
views
Give an example of a star-Menger space which is not star-$K$-Menger
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
- 505
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
3
votes
0
answers
167
views
What is the name of the class of topological spaces with the following property ....?
What is the name of the class of topological spaces with the following property $P$ ?
$X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
- 1,101
3
votes
0
answers
83
views
Does there exist a regular $P$-space which is strongly star-Lindelof but not star-Menger?
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
- 505
2
votes
1
answer
167
views
Is $\mathbb R$ with cocountable topology star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 505
2
votes
1
answer
167
views
Definition of $S_1(A,B)$
The definition of first selection principle is well known: $S_1(A,B)$.
Let $A$ and $B$ be families of sets. The symbol $S_1(A, B)$ denotes the statement:
for each sequence $(A_n : n \in \omega)$ of ...
- 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
1
vote
1
answer
149
views
Does there exist a star-Lindelöf space which is not DCCC?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V$ of $\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal ...
- 505
1
vote
1
answer
117
views
Does there exist a starcompact space which is not star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 505
1
vote
0
answers
70
views
A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz
An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite.
A space $X$ is said to be ...
- 505
0
votes
1
answer
170
views
P-filter property?
Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...
- 1,101
0
votes
1
answer
163
views
Is there a Tychonoff space $X$ such that ....?
$X$ is not a separable submetrizable, i.e.($iw(X)>\omega$) $X$ has not a countable injective weight.
There is a Baire isomorphism 1-class between $X$ and a separable metrizable space $Y$.
- 1,101