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arXiv · 2606.02040

Generalized almost disjoint families and injective Banach spaces

Chris Lambie-Hanson·David Schrittesser
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Abstract

A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an \emph{almost disjoint family} on a topological space $X$, generalizing the classical notion of almost disjoint families of subsets of $\mathbb{N}$, which we feel is of interest in its own right. We prove that, if $\mathfrak{b} = 2^{\aleph_0}$, then there exists an almost disjoint family of cardinality $2^{\aleph_1}$ on the Čech-Stone remainder of $\mathbb{N}$.

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Chris Lambie-Hanson, David Schrittesser. 2026-06-01. Generalized almost disjoint families and injective Banach spaces. https://arxiv.org/abs/2606.02040

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