Generalized almost disjoint families and injective Banach spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918535042695168 |
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| author | Lambie-Hanson, Chris Schrittesser, David |
| author_facet | Lambie-Hanson, Chris Schrittesser, David |
| contents | 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 $\mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02040 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalized almost disjoint families and injective Banach spaces Lambie-Hanson, Chris Schrittesser, David Functional Analysis Logic 03E05, 03E50, 46M10. 46M15, 46M18 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 $\mathbb{N}$. |
| title | Generalized almost disjoint families and injective Banach spaces |
| topic | Functional Analysis Logic 03E05, 03E50, 46M10. 46M15, 46M18 |
| url | https://arxiv.org/abs/2606.02040 |