# Questions tagged [selection-principles]

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45
questions

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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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118 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 ...

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232 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 ...

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70 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 ...

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59 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 ...

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88 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 ...

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120 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 ...

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232 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 ...

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50 views

### Let $X$ be a Lindelöf space. Does the following condition imply that $X$ is star-Menger?

Let $X$ be a Lindelöf space. If every cover $\{G_M : M\in\mathcal{M}(X)\}$ of $X$, where $G_M$ is a $G_\delta$ subset of $X$ with $M\subseteq G_M$, has a subcover of cardinality $\mathfrak{b}$, then ...

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128 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}\...

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139 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\...

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116 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{...

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65 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, ...

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22 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},\...

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63 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\...

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180 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 ...

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114 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 ...

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72 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 ...

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152 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 ...

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258 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 ...

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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 ...

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710 views

### Dual Borel conjecture in Laver's model

A set $X\subseteq 2^\omega$
of reals is of strong measure zero (smz) if $X+M\not=2^\omega$
for every meager set $M$. (This is a theorem of Galvin-Mycielski-Solovay,
but for the question I am going to ...

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**1**answer

267 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\...

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167 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 ...

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223 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\...

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703 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 ...

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157 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 $Y$.

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189 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 $F$ ...

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158 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 ...

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301 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 ...

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203 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 ...

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200 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 ...

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402 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 ...

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272 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 $\...

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607 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 ...

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882 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 ...

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**1**answer

347 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 ...

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433 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 ...

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### The complexity of Max-K interval selection

I came up with the following problem, but do not know how to analyze it.
Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in $...

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264 views

### Countable vs. ultra-negligible sets [duplicate]

A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us ...

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1k views

### A set that can be covered by arbitrarily small intervals

Let $X$ be a subset of the real line and $S=\{s_i\}$ an infinite sequence of positive numbers. Let me say that $X$ is $S$-small if there is a collection $\{I_i\}$ of intervals such that the length of ...

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696 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 $X$...

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249 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,...

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425 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 ...

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89 views

### Is the covering property $\Omega \choose \text{T}$ closed under products?

Suppose that $(X_i)_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see this post).
Does $\prod_{i\in I} X_i$ ...