Questions tagged [cardinal-characteristics]

For questions about various cardinal invariants, cardinal characteristics of the continuum and related topics.

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119 views

Is existence of a cardinal that witness non-failure of GCH everywhere everyway, a theorem of ZF?

In an earlier positing to $\mathcal MO$, it appears that the answer to if the $\sf GCH$ can fail everywhere in every way is to the negative, this is the case in $\sf ZFC$, however it also appears that ...
2
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0answers
69 views

A convex version of the small uncountable cardinal $\mathfrak b$

Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$. The definition of $\mathfrak ...
2
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1answer
105 views

Game versions of the tower number $\mathfrak t$

Let us recall that $\mathfrak t$ is the smallest cardinal $\kappa$ for which there exists a family $(T_\alpha)_{\alpha\in\kappa}$ of infinite subsets of $\omega$ such that $\bullet$ for any ordinals $\...
5
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1answer
156 views

The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system

I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum. For a compact ...
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111 views

Is this bi-hierarchy anti-Cantorian system consistent with AC?

In a posting to MO I've presented a bi-hierarchy anti-Cantorian system. The consistency of this system is not settled yet. However, if we assume that it is consistent, then would it be consistent with ...
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1answer
141 views

Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that $$\...
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1answer
133 views

Bounds for a covering number of the circle group $\mathbb T$ by some its small subgroups

$\newcommand{\w}{\omega}\newcommand{\A}{\mathcal A}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}...
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1answer
265 views

Bounds for a small cardinal

$\newcommand{\w}{\omega}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}{\mathfrak x}\newcommand{\...
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2answers
580 views

Diagonalizing against $\omega_1$-sequences of functions mod finite

The following statement is a direct consequence of the Continuum Hypothesis: There exists a sequence $\langle f_\alpha:\omega_1\rightarrow\omega_1 ~ \vert ~ \alpha<\omega_1\rangle$ of functions ...
12
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1answer
630 views

Can we separate the almost-disjointness sunflower numbers?

This question concerns a new cardinal characteristic of the continuum that arose out of issues in my answer to the question, Sunflowers in maximal almost disjoint families. A family $\cal A$ of ...
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167 views

How does "spreading-with-determinacy" compare with Cichon's diagram?

For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say that $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that $F[A]\in\mathbb{B}$...
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232 views

Do escaping sets "uniformly" cover dominating sets under determinacy?

For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that for all $X\in\mathbb{A}$...
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82 views

Are there results on cardinal function using o-tightness?

Recall that a space $X$ has countable $o$-tightness, if for every family $\mathcal U$ of open sets of $X$ and for each $x \in X$ with $x \in \overline{\bigcup \mathcal U}$, there exists a countable ...
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137 views

A model where the uniformity of the meager ideal is strictly below the almost disjointness number

I'm looking for a model satisfying the inequality described in the title. Recall that the uniformity of the meager ideal, denoted $\operatorname{non}(\mathcal M)$ (or $\operatorname{non}(\mathcal B)$) ...
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153 views

The list reaping number?

My question is inspired by a question of Dominic van der Zypen. Let $[\omega]^\omega$ denote the set of all infinite subsets of $\omega$. The reaping number, denoted by $\mathfrak r$, is the minimum ...
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1answer
164 views

A sequence of cardinal characteristics constructed with hypergraph coloring

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A coloring is a map $c: ...
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161 views

What is the smallest number of nowhere dense affine subsets covering a topological group?

$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$. Given a non-discrete topological ...
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125 views

Two cardinal characteristics of the continuum, related to the Bohr topology on integers

For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...
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1answer
131 views

Upper bound for constructibility orders of elements of the continuum

In a constructible universe (ZFC + V=L), is there any known upper bound for the constructibility orders of all elements of the continuum, i.e. some separately described ordinal $\alpha$ such that we ...
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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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103 views

Cardinality of the closure of subset of a dense subset

Is the following true? Given a first countable Hausdorff space $X$ and a dense subset $Y\subset X$ which is initially $2^\omega$-initially Lindelöf in $X$. Let $F$ be a subset of $Y$ such that $\vert ...
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149 views

Selecting an almost disjoint family in a given family of sets

A family $\mathcal A$ of infinite subsets of $\omega$ is called almost disjoint if for any distinct sets $A,B\in\mathcal A$ the intersection $A\cap B$ is finite. Let $\mathfrak a'$ be the largest ...
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1answer
109 views

What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$?

Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals ...
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1answer
159 views

What is the smallest cardinality of a maximal ultrafamily of infinite subsets of $\omega$?

A family $\mathcal U$ of infinite subsets of $\omega$ is called an ultrafamily if for any sets $U,V\in\mathcal U$ one of the sets $U\setminus V$, $U\cap V$ or $V\setminus U$ is finite. By the ...
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221 views

Cardinal numbers and the Bolzano-Weierstrass theorem

Let $\kappa$ be a cardinal number, define $\textsf{M}(\kappa)$ and $\textsf{BW}(\kappa)$ as follows: $\textsf{M}(\kappa)$ : For every sequence $(f_{n}:\kappa\to \mathbb{R})_{n\in\mathbb{N}}$ of real-...
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196 views

Influence of cardinal characteristics on nonstandard analysis?

As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but ...
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132 views

Strong chains of uncountable functions and cardinal characteristics

A family of functions $\langle f_\alpha:\alpha<\kappa\rangle$ from $\omega_1$ to $\omega_1$ is called a strong chain if $\alpha<\beta<\kappa\Longrightarrow \{\xi<\omega_1: f_\beta(\xi)\leq ...
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1answer
644 views

Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?

A function $f:\omega\to\omega$ is called $\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$; $\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite. Let us ...
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1answer
321 views

Calculate the $\downarrow$, $\downarrow\uparrow$ and $\uparrow\downarrow$ cofinalities of the poset of nontrivial finitary partitions of $\omega$

Let $(P,\le)$ be a poset. For a point $x\in P$ let $${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $...
7
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1answer
258 views

Pseudo-intersections, splitting families, and ultrafilters

Suppose $U$ is a non-principal ultrafilter on $\omega$, and let us define $\tau(U)$ to be the minimum cardinality of a family $\mathcal{X}\subseteq U$ such that $\mathcal{X}$ does not have an infinite ...
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1answer
95 views

Bounding and dominating numbers for preordering $\leq^*$ on bijections $f:\omega\to\omega$

For functions $f, g:\omega\to\omega$ we write $f \leq^* g$ if $\{x\in\omega: f(x)> g(x)\}$ is finite. Let $S_\omega$ denote the collection of bijections $\varphi:\omega\to\omega$ Similarly to the ...
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1answer
156 views

Bounding and domination numbers for relation $\leq$ modulo $\omega$-nullsets

We say that $A\subseteq \omega$ is a nullset if $$\lim\sup_{n\to \infty} \frac{|A\cap n|}{n+1} = 0.$$ Let $\omega^\omega$ denote the set of functions $f:\omega\to\omega$. We define a pre-ordering ...
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1answer
82 views

Thinning directed sets ${\frak P}$ of partitions of $\omega$ with no ${\frak P}$-discrete subsets

This question branches from Taras Banakh's recent question on a cardinal characteristic connected to families of partitions that are directed in the ordering of partition refinement. A partition $\...
6
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1answer
238 views

A question on simple $P_{\aleph_2}$-points

This question is motivated by discussion surrounding this MO question. An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...
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1answer
583 views

A new cardinal characteristic (related to partitions)?

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic. By a partition ...
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1answer
86 views

Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection?

Given two set $A,B$ we write $A\subset^* B$ if the complement $A\setminus B$ is infinite. A Hausdorff gap is a transfinite family $\langle A_\alpha,B_\alpha\rangle_{\alpha\in\omega_1}$ of infinite ...
6
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1answer
168 views

Self-embeddings of uncountable total orders, 2

Let $S = (\Omega,\leq)$ be an uncountable dense total order, such that for all positive integers $m$ and all finite ordered sequences $a_1 < a_2 < \ldots < a_m$ and $b_1 < b_2 < \ldots &...
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147 views

Negation of cardinal characteristics

Looking over the various cardinal characteristics of the continuum all of them are defined by a sentence of the form: "$\mathfrak{x}$ is the least cardinality of a subset of $\omega^\omega$ that $\...
5
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1answer
200 views

Relationship between "infinitely unequal" and "eventually different"

Suppose that $\kappa$ has the property that for every family $A\subseteq\omega^{\omega}$, if $|A|<\kappa$, then there exists some $g\in\omega^{\omega}$ such that for any $f\in A, \exists^{\infty}n\;...
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1answer
395 views

Smallest size of a non-measurable set of reals

The question is pretty much the title. I'm wondering if anything is known about the smallest size $\kappa$ of a non-measurable subset of the real numbers (regarding the Lebesgue measure). Since we ...
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2answers
237 views

How to increase unbounding and dominating numbers for $(\kappa^\lambda,\leq^*)$

Let $\kappa$ be regular and $\lambda\geq\kappa$. For $f, g\in\kappa^\lambda$ say that $f\le^* g$ if the set $\{\gamma<\lambda:f(\gamma)>g(\gamma)\}$ has size less than $\kappa$. Set $\mathfrak{...
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1answer
296 views

Effect of adding one Hechler real versus adding two on the meager ideal

In "The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)" by Haim Judah and Saharon Shelah JSL Vol. 55, No. 3 (Sep., 1990), pp. 909-927 ([...
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1answer
293 views

A simple cardinal characteristic associated with $\omega^\omega$

We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^*$ on $\omega^\omega$ defined by $x\leq^* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $...
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1answer
586 views

What if $\mathbb{R}$ is in bijection with the cardinals less than $\frak{c}$?

I was wondering whether it is consistent to have $\frak{c} = \aleph_{\frak{c}}$ where $\frak{c} = 2^{\aleph_0}$ is the cardinality of the reals (over ZFC). If so, what interesting consequences of this ...
7
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1answer
230 views

What's the minimal weight of a maximal space?

A non-empty topological space without isolated points is called maximal if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple ...
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2answers
341 views

Relations between two tower numbers

A tower is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has ...
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115 views

Two small uncountable cardinals related to Q-sets

A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$. Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
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2answers
446 views

Small uncountable cardinals related to $\sigma$-continuity

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
5
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1answer
470 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
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454 views

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals: $\mathfrak p$ is the ...