Questions tagged [selection-principles]

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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
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6 votes
1 answer
138 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
3 votes
1 answer
171 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
5 votes
1 answer
112 views

Is there an $\varepsilon$-space which is not $k$-Lindelöf?

Crossposted from https://math.stackexchange.com/questions/4717613 An $\omega$-cover $\mathscr U$ of a space $X$ is a collection of open sets so that $X \not\in\mathscr U$ and every finite subset of $...
C. Caruvana's user avatar
1 vote
0 answers
68 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 ...
Nur Alam's user avatar
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6 votes
1 answer
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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 $...
Nur Alam's user avatar
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3 votes
1 answer
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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 $...
Nur Alam's user avatar
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5 votes
1 answer
130 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
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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
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0 answers
155 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
4 votes
1 answer
307 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
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5 votes
1 answer
364 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
3 votes
0 answers
164 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 $V$ ...
Alexander Osipov's user avatar
3 votes
1 answer
90 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
0 answers
124 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
138 views

On the Menger property and the Alexandroff duplicate

Recall that a space $X$ is Menger if for each sequence $(\mathcal{U}_n)_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)_{n\in\omega}$ such that, for each $n\in \omega$, $\...
J. Casas's user avatar
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2 votes
1 answer
70 views

Why are the selection principle $S_\text{fin}(\Lambda, \Omega)$ and $S_\text{fin}(\mathcal{O},\Lambda)$ impossible for nontrivial spaces?

Recall that an open cover $\mathcal{U}$ of $X$ is a $\gamma$-cover if it is infinite and each $x\in X$ belongs to all but finitely many elements of $\mathcal{U}$ and an open open cover $\mathcal{V}$ ...
user494948's user avatar
1 vote
1 answer
147 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 ...
Nur Alam's user avatar
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2 votes
1 answer
155 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 $X$ ...
Nur Alam's user avatar
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1 vote
1 answer
112 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 $X$ ...
Nur Alam's user avatar
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1 vote
0 answers
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selection theory for normal non-paracompact domains?

Are there theorems in selection theory without either paracompactness or convexity assumptions ? That is, a theorem that claims existence of selections for any (perfectly or hereditary) normal spaces, ...
user420620's user avatar
3 votes
1 answer
252 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 ...
Nur Alam's user avatar
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3 votes
0 answers
82 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 ...
Nur Alam's user avatar
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8 votes
1 answer
266 views

When does $C(X)$, $X$ a continuum, admit a continuous choice function?

Given a continuum $X$ (compact metrizable connected $X$) let $K(X)$ denote the hyperspace of nonempty compact subspaces of $X$ with the Vietoris topology and let $C(X)$ denote the (closed) subspace of ...
Alessandro Codenotti's user avatar
1 vote
1 answer
80 views

Does there exist a strongly star-Lindelöf space which is not DCCC?

A space $X$ is said to be strongly star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $A$ of $X$ such that $St(A,\mathcal U)=X$. A space $X$ has discrete ...
Nur Alam's user avatar
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1 vote
1 answer
99 views

$L^p$-barycenters via continuous selectors

Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...
ABIM's user avatar
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0 votes
0 answers
210 views

k-secretary problem: not knowing the length of the queue

The secretary problem is a famous and old problem. You can find the basic definition of this problem here: https://en.wikipedia.org/wiki/Secretary_problem Now I'm concerned with the k-secretary ...
m0ss's user avatar
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8 votes
1 answer
280 views

Isotopies, Fiber Bundles and Selection Theorems

The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with ...
John Samples's user avatar
1 vote
1 answer
147 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
0 votes
2 answers
192 views

Give an example of a Rothberger space $X$ which has a Lindelöf subspace $Y$ that is not Rothberger

A space $X$ is said to be Rothberger if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(U_n)$ such that for each $n$ $U_n\in\mathcal{U}_n$ and $\{U_n : n\in\mathbb{...
Nur Alam's user avatar
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-1 votes
1 answer
92 views

Existence of continuous selection for metric projection

Let $(X,d)$ be a separable complete geodesic metric space and let $K$ be a compact (non-empty) subset of $X$. Without assuming things like linearity, the convexity of $K$, and locally convexity, ...
Catologist_who_flies_on_Monday's user avatar
2 votes
0 answers
36 views

Characterization of Menger property using Menger game

Hurewicz proved the following result : The Player ONE does not have a winning strategy in the game $G_{fin}(\mathcal{O},\mathcal{O})$ or Menger game if and only if $X$ satisfies $S_{fin}(\mathcal{O},\...
Manoj Bhardwaj's user avatar
3 votes
0 answers
67 views

Non-linear weak*-continuous left inverses

Let $T\colon X\to Y$ be a continuous linear surjection between Frechet spaces. Then it is possible to use Michael's selection theorem to show that there exists a continuous (non-linear) function $g\...
T. Milva's user avatar
3 votes
1 answer
224 views

Measurable selection for argmin to distance

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with ...
ABIM's user avatar
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3 votes
1 answer
75 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
ABIM's user avatar
  • 4,969
7 votes
1 answer
160 views

A selection principle in measure theory

A Borel subset $B$ of the unit interval $\mathbb I=(0,1)$ is defined to be a density neighborhood of a set $A\subseteq\mathbb I$ if for every $a\in A$ we have $$\lim_{\varepsilon\to0}\frac{\lambda(B\...
Taras Banakh's user avatar
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7 votes
0 answers
230 views

Does there exist a complete metric space which is Rothberger (or Menger) but not Hurewicz?

A topological space $X$ is said to be a Menger space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a ...
Nur Alam's user avatar
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0 votes
1 answer
169 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 ...
Alexander Osipov's user avatar
9 votes
2 answers
278 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 ...
Alexander Osipov's user avatar
1 vote
1 answer
185 views

Countable union of Menger spaces

A topological space $X$ has Menger's property $\textsf{S}_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}_1, \mathcal{U}_2, \cdots $, we can select finite ...
Gabriel Medina's user avatar
9 votes
1 answer
252 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\...
Taras Banakh's user avatar
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6 votes
1 answer
748 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 ...
Boaz Tsaban's user avatar
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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 $Y$.
Alexander Osipov's user avatar
7 votes
1 answer
204 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
2 votes
1 answer
165 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 ...
Alexander Osipov's user avatar
14 votes
1 answer
288 views

Strictly Fréchet spaces versus strongly Fréchet spaces

For a topological space $X$ and a point $x\in X$, consider the following definitions: (Gerlits and Nagy): $X$ is strictly Fréchet at $x\in X$ if for any sequence $(A_n)_{n\in\omega}$ such that $x\in\...
Renan Mezabarba's user avatar
8 votes
1 answer
229 views

On a strengthening of strong measure zero

Recall that a set of $X$ of reals has strong measure zero (SMZ) if for every sequence $\{\epsilon_n:n<\omega\}$ of positive real numbers, there is a sequence $\{I_n:n<\omega\}$ of intervals such ...
Todd Eisworth's user avatar
3 votes
1 answer
206 views

Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$

A $\kappa$-tower in $\mathbb{N}^\mathbb{N}$ is a sequence $\langle a_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$ that is $\le^*$-increasing with $\alpha$ and has no $\le^*$-upper ...
Boaz Tsaban's user avatar
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5 votes
0 answers
314 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 ...
Boaz Tsaban's user avatar
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5 votes
1 answer
415 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 ...
Boaz Tsaban's user avatar
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