Questions tagged [selection-principles]

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32
votes
1answer
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 ...
24
votes
2answers
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 ...
18
votes
1answer
667 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 $X$...
18
votes
2answers
696 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 ...
15
votes
2answers
849 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 ...
13
votes
1answer
255 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\...
9
votes
2answers
252 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 ...
9
votes
1answer
202 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\...
8
votes
1answer
201 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 ...
8
votes
1answer
412 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 ...
7
votes
1answer
669 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 ...
7
votes
1answer
188 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 $F$ ...
7
votes
1answer
256 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 $\...
7
votes
0answers
151 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 ...
6
votes
3answers
571 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 ...
6
votes
1answer
338 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 ...
5
votes
1answer
78 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 ...
5
votes
1answer
387 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 ...
5
votes
0answers
296 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 ...
4
votes
1answer
392 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 ...
4
votes
1answer
63 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 ...
4
votes
0answers
70 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\...
3
votes
1answer
195 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 ...
3
votes
1answer
248 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 ...
3
votes
1answer
236 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,...
3
votes
0answers
61 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\...
2
votes
1answer
153 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 ...
2
votes
1answer
86 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 $\prod_{i\in I} X_i$ ...
1
vote
1answer
130 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 ...
1
vote
0answers
23 views

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 $...
0
votes
1answer
146 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 ...
0
votes
1answer
155 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 $Y$.