Questions tagged [cardinal-characteristics]

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

Filter by
Sorted by
Tagged with
3
votes
1answer
130 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\;...
2
votes
0answers
176 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 ...
3
votes
2answers
216 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{...
9
votes
1answer
254 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 ([...
12
votes
1answer
253 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 $...
6
votes
1answer
542 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 ...
6
votes
1answer
197 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 ...
6
votes
1answer
157 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 ...
2
votes
0answers
109 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-...
9
votes
2answers
418 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
votes
1answer
417 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 ...
10
votes
0answers
445 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 ...
14
votes
0answers
494 views

Small cardinals related to topological convergence

Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
28
votes
1answer
971 views

Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable ...
10
votes
2answers
377 views

The “strong” measure number

Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, ...
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 ...
13
votes
2answers
788 views

Consistency Results Separating Three Cardinal Characteristics Simultaneously

(For information on cardinal characteristics of the continuum aka cardinal invariants see Joel David Hamkins' MO answer here; Andreas Blass's handbook article is an excellent reference.) Problem 2.3 ...
5
votes
2answers
1k views

Improvements of the Baire Category Theorem under (not CH)?

The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of complete metric ...