# Questions tagged [selection-principles]

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60
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### 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 ...

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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 $...

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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 $...

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### 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 ...

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### 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 $...

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130
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### 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 ...

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### 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 $X$ ...

5
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257
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### 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 ...

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### 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 $V$ ...

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### 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. ...

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### 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\...

3
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108
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### 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 $n\in \omega$, $\...

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### 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 $\mathcal{V}$ ...

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144
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### 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 ...

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### 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 $X$ ...

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1
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108
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### 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 $X$ ...

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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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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1
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### $L^p$-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 ...

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### 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 ...

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261
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### 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 ...

1
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1
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144
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### 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}\...

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### 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{...

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### 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, ...

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### 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},\...

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### 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\...

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163
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### 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 ...

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### 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 ...

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### 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\...

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### 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 ...

0
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1
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157
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### 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 ...

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273
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### 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
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1
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179
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### 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 ...

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242
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### 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\...

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### 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 ...

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### 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 $Y$.

7
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### 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 $F$ ...

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### 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 ...

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1
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274
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### 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\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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285
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### 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 $\...

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2
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### $\mathfrak{ufo}$: 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 ...

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### 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 ...

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### 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 ...