# Questions tagged [cardinal-characteristics]

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

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### Can the nowhere dense sets be more complicated than the meager sets?

Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
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### "Compactness length" of Baire space

Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton? In more ...
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### Is there any class of ideals for which $\mathfrak{b}(\mathcal I)\neq\mathfrak{b}$?

Consider the Baire space $\mathbb{N}^\mathbb{N}$. A natural pre-order $\leq^*$ on $\mathbb{N}^\mathbb{N}$ is defined by $f\leq^*g$ if and only if $f(n)\leq g(n)$ for all but finitely many $n$. A ...
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### Comparing bornologies for cardinal characteristics via Borel maps

This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
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### Complemented subspaces of a dual Banach space

Let $\kappa$ be an infinite cardinal number and by $\mathcal{B}(\kappa)$ denote the class of all Banach spaces of density $\kappa$. My question reads as follows: Does there exist $\kappa$ for which ...
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Suppose $(P, <)$ is a poset of cofinality $\aleph_2$ and additivity (least cardinality of an unbounded subset) $\aleph_1$. Can we conclude the existence of a cofinal subset of order-type $\omega_1 \... 7 votes 1 answer 225 views ###$\operatorname{cof}(\mathcal{L}) = \aleph_1$and$2^{\aleph_0} = \aleph_3$Here$\operatorname{cof}(\mathcal{L})$refers to the largest cardinal characteristics of the continuum in Cichon's diagram. My question is: Is the theory$\mathsf{ZFC}+ \operatorname{cof}(\mathcal{L})...
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I have come across the following notion of (what I am calling) dual ideals, and I am looking for any work in which this notion has been considered, and particularly anything about transferring ...
1 vote
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### 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 ...
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### 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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### 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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### 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 $X\in\mathbb{A}$...
1 vote
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### 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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### 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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### 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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### 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 ...
### 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 ...